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Class 7 Mathematics Chapter 12 of 15

Chapter 12 — Algebraic Expressions

Overview

This chapter introduces Algebraic Expressions to Class 7 students from the CBSE textbook Mathematics – VII. It begins with the concept of variables and constants, and shows how letters are used to represent numbers in expressions. The chapter develops skills to form, simplify and manipulate expressions: identifying terms, factors and coefficients; classifying expressions as monomials, binomials, trinomials and polynomials; finding degrees; and performing addition, subtraction and multiplication of algebraic expressions. Emphasis is placed on evaluating expressions by substitution, combining like terms, taking common factors (basic factorisation) and applying algebra to model and solve simple word problems. Learning these ideas builds algebraic thinking needed for higher classes and helps translate real-life situations into mathematical form.

Learning Objectives

  • Define algebraic expression, term, factor, coefficient and constant
  • Identify monomials, binomials, trinomials and polynomials from given expressions
  • Classify algebraic expressions by degree and by number of terms
  • Differentiate between like terms and unlike terms and recognise them in expressions
  • Combine like terms to simplify algebraic expressions
  • Apply the distributive property to remove brackets and expand expressions
  • Perform addition and subtraction of algebraic expressions
  • Evaluate algebraic expressions by substituting given numerical values for variables

Topics in this chapter

11 topics · tap a topic title to jump straight to it.

🔣1

Introduction to Algebraic Expressions

What is an algebraic expression? An algebraic expression is a combination of numbers, variables (letters like x, y), and arithmetic operations (addition, subtraction, multiplication, division, powers) but it does not contain an equals sign. Example: 3x + 5, 2a^2 - 4a + 7.

Parts of an expression

  • Variable: a letter that represents an unknown number (e.g., x, y).
  • Coefficient: the numerical factor multiplied by the variable (in 5x, 5 is the coefficient).
  • Constant: a number without a variable (e.g., 7).
  • Term: each part of the expression separated by + or − (e.g., in 3x + 4y − 2, terms are 3x, 4y, −2).

Types of expressions

  • Monomial: one term (e.g., 7, 3x, 2a^2).
  • Binomial: two terms (e.g., x + 5, 3x − 2y).
  • Polynomial: one or more terms (e.g., 2x^2 + 3x + 1).

Like and unlike terms

  • Like terms have the same variable part and same powers (e.g., 3x and −5x are like terms).
  • Unlike terms have different variable parts or powers (e.g., 2x and 3x^2 are unlike).

Simplifying expressions: combine like terms using addition or subtraction and use distributive property to remove parentheses. Example: 3x + 5x − 2 = 8x − 2. For 2(x + 3), use distributive property: 2x + 6.

Degree: the degree of a term is the power of the variable in that term (e.g., degree of 4x^2 is 2). The degree of a polynomial is the highest degree among its terms (e.g., degree of 2x^2 + 3x + 1 is 2).

Evaluation: to evaluate an expression, substitute a number for each variable and compute. Example: if x = 2, evaluate 3x + 5 → 3(2) + 5 = 11.

Difference between expression and equation: an expression is a mathematical phrase without an equals sign; an equation has an equals sign and states that two expressions are equal (e.g., 3x + 5 = 11).

How algebraic expressions help: they let us model real situations symbolically so we can form general rules and solve problems without repeating computation for every case.

📌 Examples
  • 1) Identify parts: In 5x^2 − 3x + 8: terms = 5x^2, −3x, 8; variables = x; coefficients = 5, −3; constant = 8; degree = 2.
  • 2) Simplify: 4x + 7 − 2x + 3 = (4x − 2x) + (7 + 3) = 2x + 10.
  • 3) Use distributive property: 3(2x − 4) = 6x − 12.
  • 4) Evaluate: For x = 3, evaluate 2x^2 + x − 5 → 2(3)^2 + 3 − 5 = 18 + 3 − 5 = 16.
  • 5) Word problem: If one notebook costs Rs. x and you buy 4 notebooks and a pen costing Rs. 15, total cost = 4x + 15. If x = 30, total = 4(30) + 15 = 135.
🧮 Formulas
  1. Term types: Monomial (one term), Binomial (two terms), Polynomial (one or more terms).
  2. Like terms: same variables with same powers (combine by adding coefficients). Example: ax + bx = (a + b)x.
  3. Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac.
  4. Degree of a term: exponent of variable (e.g., degree of 7x^3 is 3). Degree of polynomial: highest term degree.
  5. Evaluation: substitute value for variable. Example: For x = n, evaluate P(x) by replacing x with n.
  6. Combining terms: 3x + 5x = (3 + 5)x = 8x; 2x^2 + 3x − x = 2x^2 + 2x.
📊 Visual ideas
Plot a straight line y = 2x + 3 to visualize coefficient (slope = 2) and constant (y-intercept = 3). Use a table of x values (−2, −1, 0, 1, 2) and plot corresponding y.
Plot y = x to show a simple linear relationship (slope 1, intercept 0). Color the line and label points to link each point to the value of the expression 'x'.
Use bar or column visuals to represent terms: for example, draw two bars for 3x and 5x and show combining them into one bar labeled 8x (helps to visualize combining like terms).
Plot values of a quadratic expression like y = x^2 + 2x + 1 for integer x from −3 to 3 to show how degree 2 polynomials curve (optional extension).
🔢2

Terms, Factors and Coefficients

Overview
In algebraic expressions, understanding terms, factors and coefficients helps you read, simplify and manipulate expressions.

Terms
A term is a single part of an expression separated by plus (+) or minus (−) signs. Examples of terms: 5x, −3xy, 7, 4a^2. Terms can be monomial (one term), binomial (two terms) or trinomial (three terms). A constant term is a term without any variable (e.g., 7).

Like and unlike terms
Like terms have exactly the same variable part (same variables raised to the same powers). For example, 3x^2y and −5x^2y are like terms; 2x and 2x^2 are unlike. Only like terms can be added or subtracted by combining their coefficients.

Coefficients
The coefficient of a term is the numerical factor that multiplies the variable part. In 7ab^2, 7 is the coefficient. If no number is written, the coefficient is 1 (for x, coefficient = 1; for −y, coefficient = −1).

Factors
Factors are quantities being multiplied. In a term like 6xy, 6, x and y are factors. In factorisation you write an expression as a product of factors, for example 6x + 9 = 3(2x + 3) — here 3 is the common factor. The distributive law connects sums and factors: a(b + c) = ab + ac.

How to identify and work with them

  • To list terms: split the expression at + and − signs (keeping sign with each term).
  • To find coefficient: look at the number multiplying the variable part.
  • To combine like terms: add/subtract their coefficients and keep the common variable part.
  • To factor out a common factor: find the greatest common factor (GCF) of coefficients and variables, then use distributive law.

📌 Examples
  • Expression: 5x^2 − 3xy + 7. Terms: 5x^2, −3xy, 7. Coefficients: 5, −3, (constant 7 has no variable).
  • Combine like terms: 4x + 3 − 2x + 6 = (4x − 2x) + (3 + 6) = 2x + 9.
  • Factor common factor: 8ab + 12a = 4a(2b + 3). Here factors of each term include 4 and a; coefficient 4a is factored out.
  • Identify like/unlike terms: 2x^2y and −7xy^2 are unlike (powers of x and y differ).
  • Implicit coefficient 1 and −1: x = 1·x (coefficient 1), −y = (−1)·y (coefficient −1).
  • Real-life: Cost expression — If one pencil costs r rupees, cost of 5 pencils is 5r (term: 5r; coefficient 5; factor r).
🧮 Formulas
  1. General term: a·x^n — here a is the coefficient, x^n is the variable part (power n).
  2. Distributive law (connects factors and terms): a(b + c) = ab + ac.
  3. Combine like terms: (m·x^k) + (n·x^k) = (m + n)·x^k (add coefficients m and n).
  4. Factorisation of common factor: ax + ay = a(x + y).
  5. Degree of a term: sum of exponents of variables in the term (e.g., degree of 3x^2y is 2 + 1 = 3).
  6. Zero coefficient: 0·x = 0 (term vanishes if coefficient is 0).
📊 Visual ideas
Color-coded term bars: For the expression 3x + 5 − 2x, draw three horizontal bars (3x, 5, −2x) with different colors; combine the x-bars (3x and −2x) visually to show result 1x + 5. Label each bar with its coefficient and variable.
Algebra tiles / rectangle model: Use unit tiles for constants and variable tiles for x to show 4x + 6 factoring as 2(2x + 3). Represent grouping of tiles into equal rows (factors) to find common factor.
Factor-rectangle for binomials: For ax + ay draw a rectangle with side a and side (x + y) to illustrate a(x + y) = ax + ay; split rectangle to show two terms.
Line graph to show coefficient as slope: Plot y = 3x and y = x to compare slopes (3 and 1). This shows how the coefficient of x affects steepness. Use axes labeled x and y and mark slope values.
🔣3

Types of Algebraic Expressions

Algebraic expression: A combination of numbers, variables and arithmetic operations (addition, subtraction, multiplication, division). Examples: 5x, x + 3, 2x2 + x + 1.

  • Monomial: An expression with only one term. Examples: 7, 3x, -2xy, 4a2b. Degree of a monomial = sum of exponents of its variables (degree of 4a2b is 3).
  • Binomial: An expression with two terms separated by + or −. Examples: x + 5, 3x − 2y. Degree of a binomial = the highest degree among its terms (e.g., degree of x + x2 is 2).
  • Trinomial: An expression with three terms. Example: x2 + x + 1.
  • Polynomial: An expression made of one or more terms (monomials) connected by + or −. General form: anxn + an−1xn−1 + … + a1x + a0. Degree of a polynomial = highest power of the variable with a nonzero coefficient. A nonzero constant has degree 0; the zero polynomial has no well-defined degree in elementary study.

Like terms: Terms with exactly the same variable part (same variables raised to same powers). Example: 3x and −5x are like terms. Unlike terms: Terms with different variable parts, e.g., 2x and 2x2 or x and y.

Simplifying: Combine like terms by adding/subtracting coefficients. Example: 3x + 5x − 2x = (3+5−2)x = 6x. Multiplying terms adds exponents: (2x)(3x) = 6x2.

These types help classify expressions and decide methods for simplification, evaluation and graphing.

📌 Examples
  • Monomial: 5x, -3, 7ab
  • Binomial: x + 5, 4y - 2
  • Trinomial: x^2 + x + 1
  • Polynomial: 2x^3 - x + 7
  • Like terms: 3x and -5x; 2a^2b and -7a^2b
  • Unlike terms: 2x and 2x^2; x and y
🧮 Formulas
  1. Degree of a term = sum of exponents of the variables in the term (e.g., degree of 4a^2b = 2 + 1 = 3)
  2. Degree of a polynomial = highest degree among its terms (e.g., degree of x^3 + 2x^2 + x + 5 is 3)
  3. \[General polynomial form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]
  4. Combine like terms: ax + bx = (a + b)x
  5. \[Multiplying powers: x^m · x^n = x^{m+n}\]
  6. Constant (non-zero) is a polynomial of degree 0; the zero polynomial has no defined degree in basic arithmetic
📊 Visual ideas
Plot y = 3x (monomial, linear) on graph paper or Desmos to see a straight line with slope 3.
Plot y = x^2 (monomial, quadratic) to see a parabola. Use a table of values for x = -2, -1, 0, 1, 2.
Plot y = x + 5 (binomial, linear) to observe shifting of the line up by 5 units.
Plot y = x^2 + x + 1 (trinomial, quadratic) to observe how additional terms shift and skew the parabola.
🔢4

Like and Unlike Terms

What is a term? — A term is a single part of an algebraic expression made up of a coefficient (a number) multiplied by a variable part (literal part), e.g. 3x, −5, 2xy, 4x². The coefficient is the numerical factor (3 in 3x). The literal part is the variable(s) with their powers (x in 3x, x² in 4x²).

Like terms — Terms having exactly the same literal part (same variables with the same exponents) are called like terms. Only their coefficients may differ. Example: 3x and 5x are like terms; 2x² and −7x² are like terms; 4xy and 9xy are like terms.

Unlike terms — Terms that do not have the same literal part are unlike terms. Example: 3x and 3y are unlike terms; x and x² are unlike; 2xy and 2x are unlike.

Why this matters — You can only add or subtract like terms directly by adding or subtracting their coefficients. Unlike terms cannot be combined by addition or subtraction.

How to combine like terms — step-by-step

  • 1) Identify and group terms with identical literal parts (same variables and powers).
  • 2) Add or subtract their coefficients (keep the common literal part unchanged).
  • 3) Write the result with the new coefficient followed by the common literal part. If the coefficient becomes 0, the terms cancel out.

Examples of simplification

  • 3x + 5x = (3 + 5)x = 8x
  • 6x² − 2x² = (6 − 2)x² = 4x²
  • 4xy + (−2xy) = (4 − 2)xy = 2xy
  • 2x + 3y − x + 5x = (2x − x + 5x) + 3y = 6x + 3y
  • 3x + 4y cannot be combined further because x and y are different literal parts (unlike terms).
  • 5x − 5x = 0 (the x-terms cancel out).

Notes

  • Order of variables in the literal part does not matter: 2xy and 3yx are like terms because xy = yx.
  • Exponents must match exactly: x and x² are not like terms.
  • A constant (like 7) is a term with no variable and can combine only with other constants.
📌 Examples
  • 3x + 5x = 8x (both terms have the same literal part x so we add coefficients 3 + 5 = 8).
  • 6x^2 + 2x^2 = 8x^2 (x^2 terms combine; exponents must match).
  • 4xy − xy = 3xy (4 − 1 = 3; treat xy as the common literal part).
  • 2x + 3y = cannot be simplified (x and y are unlike terms).
  • 10m − 4m + 2n = (10 − 4)m + 2n = 6m + 2n (group m terms together).
  • Real-life: If you collect 3 apples and later 5 apples, you have 8 apples (3a + 5a = 8a). If you collect 3 apples and 5 oranges (3a + 5o), you cannot combine them because apples and oranges are unlike 'terms'.
🧮 Formulas
  1. Definition of like terms: Terms with identical literal parts (same variables with same exponents).
  2. Combine like terms: ax + bx = (a + b)x
  3. Combine like terms (subtraction): ax − bx = (a − b)x
  4. Higher powers: ax^n + bx^n = (a + b)x^n (exponent n must be same)
  5. Multiple variables: a·x·y + b·x·y = (a + b)xy
  6. Zero result: ax − ax = 0
📊 Visual ideas
Bar-group visualization: Draw two color-coded bars for each term type (e.g., blue bars for x-terms, red bars for y-terms). Stack bars representing coefficients (3 blue blocks + 5 blue blocks → one taller blue bar of height 8). This visually shows combining like terms and why different colors (x vs y) don't merge.
Algebra tiles diagram: Use rectangular tiles for variable terms and small squares for units. Show 3 'x' tiles and 5 'x' tiles grouped together to form 8 'x' tiles. Show that 'x' tiles and 'y' tiles remain separate.
Flowchart: A simple flow chart that starts with an expression, branches to 'same literal part?' If yes → add/subtract coefficients; if no → keep separate. Useful as a classroom poster.
Number-line or stacked blocks for coefficients: Represent coefficients on a number line or stack positive/negative blocks to show cancellation (e.g., 5x and −5x cancel to 0).
🔣5

Simplification of Algebraic Expressions

What is simplification? Simplification of an algebraic expression means rewriting it in a simpler or more compact form without changing its value. The main tasks are removing parentheses, using properties of operations, and combining like terms.

Key ideas

  • Terms: parts of an expression separated by + or − (for example 3x, −5, 2ab).
  • Like terms: terms with the same variable(s) raised to the same power (e.g., 4x and −7x are like terms; 3x and 3x^2 are not).
  • Coefficient: the numerical factor of a term (in 5x, 5 is the coefficient).

Steps to simplify

  1. Use BODMAS (brackets, orders, division, multiplication, addition, subtraction): simplify inside brackets first.
  2. Remove brackets using the distributive property: a(b + c) = ab + ac and −(a + b) = −a − b.
  3. Collect and add/subtract like terms: ax + bx = (a + b)x.
  4. Factor common numerical or algebraic factors if required (for a simpler factored form).

Examples inside explanation

  • Simplify 3x + 5x − 2x: combine like terms → (3 + 5 − 2)x = 6x.
  • Simplify 4(a + 2) − 2(a − 3): expand → 4a + 8 − 2a + 6 → (4a − 2a) + (8 + 6) = 2a + 14.

Tips and common mistakes

  • Never combine unlike terms (e.g., x and y cannot be added into a single term).
  • Watch signs when removing brackets — a minus sign in front of brackets changes all signs inside.
  • Use color-coding or underlining to identify like terms when expressions get long.
📌 Examples
  • 1) 3x + 5x − 2x = (3 + 5 − 2)x = 6x
  • 2) 5a − (2a + 3) = 5a − 2a − 3 = 3a − 3
  • 3) 3(x + 2) + 2(x − 1) = 3x + 6 + 2x − 2 = (3x + 2x) + (6 − 2) = 5x + 4
  • 4) 2x + 3y + x − 4y = (2x + x) + (3y − 4y) = 3x − y
  • 5) 4(2m − 3) − 3(m + 5) = 8m − 12 − 3m − 15 = (8m − 3m) + (−12 − 15) = 5m − 27
  • 6) Word example: Cost of n notebooks at ₹7 each and a pen at ₹12 is 7n + 12. If you already had 2 notebooks worth, total cost for (n − 2) more is 7(n − 2) + 12 = 7n − 14 + 12 = 7n − 2
🧮 Formulas
  1. Like terms combine: ax + bx = (a + b)x
  2. Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac
  3. Remove negative bracket: −(a + b) = −a − b
  4. Commutative properties: a + b = b + a, ab = ba
  5. Associative properties: (a + b) + c = a + (b + c), (ab)c = a(bc)
  6. Multiplying coefficients: (ka)(mb) = (k·m)ab (useful when variables are same or different)
📊 Visual ideas
Bar model: draw coloured bars for each term (e.g., three red bars for 3x and two red bars for 2x) to visually combine like terms into a single bar of 5x.
Number-line scaling: represent x as a unit length; show 2x, 3x and their sum 5x along the number line to illustrate coefficients as scaling.
Linear plot: plot y = 2x + 3 and y = x + 5 on graph paper or Desmos to connect simplified linear expressions to slope (coefficient of x) and intercept (constant). Range suggestion: x from −5 to 5, y accordingly.
Bracket expansion animation: use GeoGebra or Desmos to animate a(b + c) splitting into ab and ac so students see distributive property visually.
➕6

Addition and Subtraction of Algebraic Expressions

What is an algebraic expression? An algebraic expression is a combination of numbers and letters (variables) connected by operations like +, −, ×. Example: 3x + 5, 2a − 4b.

Goal of addition and subtraction: To simplify an expression by combining like terms. Like terms are terms that have the same variable(s) raised to the same power. Only like terms can be added or subtracted.

Steps to add or subtract algebraic expressions:

  • 1. Remove parentheses (use distributive property for signs): a − (b + c) = a − b − c.
  • 2. Arrange terms and group like terms together (same variables and powers).
  • 3. Add or subtract the numerical coefficients of like terms, keeping the common variable part unchanged: 3x + 5x = (3+5)x = 8x.
  • 4. Write the result in simplified form; order terms (optional) by variables, powers, or descending order.

Important points and properties:

  • Commutative law for addition: a + b = b + a — you can reorder terms.
  • Associative law: (a + b) + c = a + (b + c) — you can regroup terms when adding.
  • Subtraction works by adding the additive inverse: a − b = a + (−b).
  • Zero and negatives: 0x = 0; adding a term and its negative cancels: 5y + (−5y) = 0.

Common mistakes to avoid:

  • Do not combine unlike terms: 3x and 2y cannot be combined.
  • Watch signs when removing parentheses, especially after a negative sign: 4a − (3a + 2b) = 4a − 3a − 2b.
📌 Examples
  • Example 1 (Addition of like terms): 3x + 5x = (3 + 5)x = 8x.
  • Example 2 (Subtraction of like terms): 7y − 2y = (7 − 2)y = 5y.
  • Example 3 (Different variables, cannot combine): 4a + 3b stays 4a + 3b because a and b are different.
  • Example 4 (Removing parentheses with minus): 4a − (3a + 2b) = 4a − 3a − 2b = a − 2b.
  • Example 5 (Combine several terms): 2x + 3y − x + 5 − 2y + 4 = (2x − x) + (3y − 2y) + (5 + 4) = x + y + 9.
🧮 Formulas
  1. Like terms combine by adding/subtracting coefficients: ax + bx = (a + b)x
  2. Subtraction as addition of inverse: A − B = A + (−B)
  3. Distributive law for removing parentheses: k(a + b) = ka + kb and a − (b + c) = a − b − c
  4. Additive identity: a + 0 = a
  5. Additive inverse: a + (−a) = 0
  6. Commutative law (addition): a + b = b + a
📊 Visual ideas
Bar-model or stacked-bar visualization: represent each term as a colored bar segment whose length equals its coefficient. To add/subtract, place bars for like-variable terms together and combine lengths (use different colors for different variables).
Number-line for coefficients: show positive and negative coefficients on a number line. Example: add −3x and 5x by showing −3 and +5 on a line and combining to +2x.
Algebra tiles diagram: use rectangular tiles for variable terms and square tiles for constants. Physically remove tile pairs that cancel (e.g., +x and −x) to demonstrate simplification.
Flowchart diagram: a simple step-by-step flow (Remove parentheses → Group like terms → Add/subtract coefficients → Write final expression) to show the process visually.
✖️7

Multiplication of a Monomial and a Polynomial

What is a monomial and a polynomial?

A monomial is an algebraic expression with a single term (for example, 3x, -4ab, 2x^2). A polynomial is a sum of one or more terms (for example, x + 3, 2x^2 - x + 5).

Rule for multiplication

To multiply a monomial by a polynomial, use the distributive law: multiply the monomial with each term of the polynomial and then add the results. When multiplying variable factors, add the exponents (for the same base).

So, if m is a monomial and P(x) = a0 + a1 x^n1 + a2 x^n2 + ... then

m * P(x) = m*a0 + m*a1 x^n1 + m*a2 x^n2 + ...

Steps

  • Multiply the numerical coefficients.
  • Multiply the variable parts: add exponents for like bases.
  • Simplify and combine like terms if any appear.

Important points

  • Multiplication is distributive: monomial*(sum) = sum of monomial*each term.
  • Negative signs distribute too: -k * ( ... ) gives negatives of each product.
  • If the monomial contains x, the resulting polynomial will have each term multiplied by that x, often increasing degrees by 1.
📌 Examples
  • 1) Multiply 3x by (2x + 5). Step 1: Distribute: 3x * 2x + 3x * 5 Step 2: Multiply coefficients and add exponents: 6x^2 + 15x Answer: 6x^2 + 15x
  • 2) Multiply (-4ab) by (3a^2 - 2b + 5). Step 1: Distribute: (-4ab)*3a^2 + (-4ab)*(-2b) + (-4ab)*5 Step 2: Multiply: -12a^3b + 8ab^2 - 20ab Answer: -12a^3b + 8ab^2 - 20ab
  • 3) Multiply 0.5x by (4x^2 - 6x + 2). Step 1: Distribute: 0.5x*4x^2 - 0.5x*6x + 0.5x*2 Step 2: Compute: 2x^3 - 3x^2 + x Answer: 2x^3 - 3x^2 + x
  • 4) Real-life: Area of a rectangular garden whose width is 3x metres and length is (2x + 3) metres. Area = width * length = 3x*(2x + 3) = 6x^2 + 9x (square metres).
🧮 Formulas
  1. Distributive law: k*(a + b + c + ...) = ka + kb + kc + ...
  2. \[General: (kx^m) * (sum_{i} a_i x^{n_i}) = sum_{i} (k*a_i) x^{m + n_i}\]
  3. \[Law of exponents: x^p * x^q = x^{p + q}\]
  4. Multiplying constants: multiply numeric coefficients normally, then attach variable part
📊 Visual ideas
Area model diagram: draw a rectangle with one side labeled 2x and the other split into x and 3 to show (2x)(x + 3). Sub-rectangles represent 2x*x and 2x*3, i.e., 2x^2 and 6x. This visualizes the distributive step.
Plot the single-variable result of an example, e.g., y = 6x^2 + 15x (from 3x*(2x+5)). Use a graphing tool (Desmos/GeoGebra). Mark the vertex and x-intercepts. Note: this is a parabola opening upward; it passes through the origin if the polynomial has factor x.
For y = 2x^3 - 3x^2 + x (from 0.5x*(4x^2 - 6x + 2)), plot to show cubic behaviour. Use different colors for original polynomial and the scaled result to show effect of multiplying by monomial.
Bar-model or flow-chart: show distribution as arrows from the monomial to each term of the polynomial, labeling each product. This helps students follow the distributive process step-by-step.
🔢8

Evaluation and Substitution

What is Evaluation and Substitution?

Evaluation of an algebraic expression means finding its numerical value when the variable(s) are given particular number(s). Substitution is the process of replacing each variable by its given value and then simplifying the resulting arithmetic expression.

Steps to evaluate an expression

  1. Identify the variable(s) and their given value(s).
  2. Substitute (replace) each variable with its value.
  3. Follow the order of operations (BODMAS/BIDMAS): Brackets, Orders (powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).
  4. Simplify by calculating powers, doing multiplications/divisions, and then additions/subtractions. Combine like terms where applicable.

Important points

  • If an expression has more than one variable, substitute values for all variables.
  • Watch signs: when substituting negative numbers, keep parentheses to avoid sign mistakes (e.g., x = -2, use ( -2 ) ).
  • Fractions and decimals can be substituted the same way—use correct arithmetic.

Why it matters

Evaluation links algebra to real life: formulas (like cost, area, perimeter) are algebraic expressions and we evaluate them to get numerical answers for specific situations.

📌 Examples
  • Example 1: Evaluate 3x + 2 when x = 4. Substitute x → 4: 3(4) + 2 = 12 + 2 = 14.
  • Example 2: Evaluate 2x^2 - 3x + 5 for x = 2. Substitute: 2(2^2) - 3(2) + 5 = 2(4) - 6 + 5 = 8 - 6 + 5 = 7.
  • Example 3: Evaluate 2x + 3y when x = 1 and y = -2. Substitute: 2(1) + 3(-2) = 2 - 6 = -4. (Use parentheses for negative values.)
  • Example 4: Evaluate 1/2 a + 3 for a = 4. Substitute: (1/2)(4) + 3 = 2 + 3 = 5.
  • Example 5: Evaluate (x - 2)(x + 3) for x = 1. Substitute: (1 - 2)(1 + 3) = (-1)(4) = -4.
🧮 Formulas
  1. Substitution rule: replace each variable by its given value, then simplify.
  2. Order of operations (BODMAS): Brackets, Orders (powers), Division/Multiplication, Addition/Subtraction.
  3. Combine like terms: ax + bx = (a + b)x.
  4. Power evaluation: x^n means multiply x by itself n times; evaluate powers before multiplication/division when following BODMAS.
  5. When substituting negatives: write the number in parentheses, e.g., x = -3 → use ( -3 ) to avoid sign errors.
📊 Visual ideas
Graph 1 (linear): Plot y = 3x + 2. Make a table of values, e.g., x = -2,-1,0,1,2 → y = -4,-1,2,5,8. Plot points and draw the straight line. This shows how substituting x gives corresponding y.
Graph 2 (quadratic): Plot y = x^2 - 2x + 1. Use x = -1,0,1,2,3 to get y = 4,1,0,1,4. Plot points to see the parabola vertex at x = 1 (evaluated visually).
Graph 3 (two variables - bar model): For a cost expression C = 50x + 30, show a bar diagram or stacked bars: one part 50x (variable cost) and one part 30 (fixed cost). Evaluate for x = 0,1,2 to show numerical totals.
Graph 4 (mapping diagram): Draw an input-output box: input value(s) → substitute into expression → output value. Example: input x=2 → evaluate 2x^2 - 3x + 5 → output 7.
👑9

Factorisation (Taking Common Factor)

What is factorisation (taking common factor)?
Factorisation by taking a common factor is the process of rewriting an expression as a product of a factor and another expression, using the distributive law. You reverse the expansion a(b + c) = ab + ac to write terms like ab + ac as a(b + c).

Why we do it
It simplifies expressions, makes solving equations easier and helps in simplifying algebraic fractions.

Key idea (Distributive law)
If each term in an expression has a common number or variable, take that common part out. Formally: ab + ac = a(b + c).

Step-by-step method

  • Look at each term and identify the greatest common factor (GCF) of the numerical coefficients.
  • For variables, take each variable that appears in every term with the smallest power.
  • Write the GCF outside parentheses and divide each term by the GCF to get the parentheses content.
  • Check by expanding: GCF × (resulting expression) should equal the original expression.

Special cases and tips

  • If all terms are negative, you can take out a negative sign as the common factor: -(a + b) = -a - b.
  • Always take the highest possible common factor (largest number and highest power of variables common to all terms).
  • When coefficients have no common number other than 1, but variables do, you may factor only the variable part (e.g., xy + xz = x(y + z)).
  • Use factoring before solving equations or simplifying fractions to cancel common factors.

How to check
Multiply (expand) the factor outside the parentheses with each term inside. If you get the original expression, factoring is correct.

📌 Examples
  • 12a + 18b = 6(2a + 3b) — common numerical factor is 6.
  • xy + xz = x(y + z) — x is common in both terms.
  • 3x^2 + 6x^3 = 3x^2(1 + 2x) — take the smallest power x^2 and factor 3.
  • -5p + 15q = 5(-p + 3q) or -(5p - 15q) = -5(p - 3q) — show extracting a positive or negative factor.
  • 8m^2n + 12mn^2 = 4mn(2m + 3n) — GCF of coefficients is 4, variables: mn (lowest powers).
  • 20x + 30 = 10(2x + 3) — factor a number from mixed numeric-variable expression.
🧮 Formulas
  1. Distributive law: a(b + c) = ab + ac (and conversely: ab + ac = a(b + c))
  2. Greatest common factor (GCF) of terms: GCF(numerical coefficients) × product of common variables with smallest powers
  3. Factoring the negative: -a(b + c) = -ab - ac (so -ab - ac = -a(b + c))
  4. Check by expansion: Factor × (parenthesis) = original expression
📊 Visual ideas
Area-model rectangle: Draw a large rectangle split into two smaller rectangles whose areas are the terms (e.g., a×b and a×c). The common side a is visible, showing ab + ac = a(b + c).
Bar model (strip diagrams): Represent each term as a bar of length proportional to the term; group the common length at one end to show the common factor outside the parentheses.
Algebra tiles/grid: Use tiles for units and x’s; arrange tiles in rows that share a common column to visually factor out the common column.
Stepwise flow diagram: Show steps in boxes — 'Find GCF' → 'Divide each term by GCF' → 'Write GCF(quotients)' → 'Check by expansion' — useful as a classroom poster.
🔣10

Forming Algebraic Expressions from Word Problems

What is an algebraic expression?
An algebraic expression is a combination of numbers, variables (like x, y), and operations (+, −, ×, ÷). In word problems we translate words into symbols to form such expressions.

Steps to form an expression from a word problem

  • Read the sentence carefully and identify the unknown quantity — assign a variable (for example, let the number be x).
  • Identify keywords that indicate mathematical operations (see keyword list below).
  • Translate each phrase into symbols, keeping the order in mind (note: phrases like “less than” reverse the order).
  • Use parentheses when the phrase groups operations (e.g., "twice the sum of a number and 3" → 2(x + 3)).
  • Combine like terms and simplify if possible.

Common keywords and their meanings

  • sum/plus/added to → +
  • difference/minus/less → − (watch the order: "3 less than x" = x − 3, but "a number less than 3" = 3 − x)
  • product/times/of → ×
  • quotient/divided by → ÷
  • twice → 2×, thrice → 3×
  • more than / less than often reverse the order ("more than" can be x + 5 or 5 + x depending on wording)
  • consecutive numbers → n, n + 1, n + 2, ...

Key points to remember

  • Assign a clear variable for the unknown(s).
  • Use parentheses to show grouped operations.
  • Check the sentence meaning to avoid reversing terms incorrectly (especially for "less than").
  • Simplify the final expression by combining like terms where possible.

Example translations (short)
"Twice a number decreased by 5" → 2x − 5
"Three less than a number" → x − 3 (but "a number less than 3" → 3 − x)
"The sum of a number and 7" → x + 7
"Twice the sum of a number and 3" → 2(x + 3)

How to check
Substitute a value for the variable (for example x = 2) into your expression and see whether the numeric result matches the meaning of the original sentence.

📌 Examples
  • 1) "Twice a number decreased by 5" → Let the number be x. Expression: 2x − 5. Check: If x = 4, expression = 2(4) − 5 = 8 − 5 = 3.
  • 2) "Three less than a number" → Let number = x. Expression: x − 3. (Compare: "a number less than 3" = 3 − x.)
  • 3) "The perimeter of a rectangle with length l and breadth b" → Perimeter = 2(l + b).
  • 4) "Total cost of n notebooks at Rs 12 each" → Cost = 12n.
  • 5) "Rahul is x years old. His sister is 4 years younger." → Sister's age = x − 4.
  • 6) "Twice the sum of a number and 3" → Let number = x. Expression: 2(x + 3) (note parentheses: not 2x + 3).
🧮 Formulas
  1. Key word → symbol: plus/sum/added to → + ; minus/difference/less → − ; times/product/twice/thrice → × ; divided by/quotient → ÷
  2. Twice a number: 2x ; Thrice a number: 3x
  3. Consecutive integers: n, n + 1, n + 2, ...
  4. Perimeter of rectangle: 2(l + b)
  5. Area of rectangle: l × b
  6. Perimeter of square (side s): 4s ; Area of square: s^2
📊 Visual ideas
Number-line visual: show a variable x and examples of expressions for different x values (e.g., mark x, x + 3, x − 2 on the same line) — helps link expressions to positions.
Bar model / strip diagram: represent parts (like 'a number' and 'add 7') as adjoining bars to visualize x + 7.
Cartesian plot of a linear expression: treat y as the expression value, e.g., y = 2x + 3. Make a table of values x = −2, −1, 0, 1, 2, compute y, plot points (x,y) and draw the straight line. This shows how the expression changes with x.
Compare two expressions on one graph: plot y1 = 2x + 3 and y2 = x − 5 to see intersections and differences in slope.
🔢11

Practice Problems and Applications

What this topic covers
The practice problems and applications section helps you use algebraic expressions to model, solve and interpret everyday situations. You learn to form expressions from words, simplify them, evaluate for given values, and use basic properties (like distributive law and combining like terms) to solve problems.

Step-by-step approach to solve word problems

  1. Read and understand: Identify what is known and what must be found.
  2. Choose a variable: Usually use x (or any letter) for the unknown.
  3. Form an expression: Translate words into symbols (e.g., "5 more than a number" → x + 5).
  4. Simplify: Combine like terms and use distributive property if needed.
  5. Evaluate or solve: Substitute values for the variable or solve an equation if one is formed.
  6. Interpret the result: Check units and whether the result makes sense in context.

Common types of problems

  • Forming expressions from word phrases (e.g., "twice a number minus 7").
  • Evaluating expressions by substituting number(s).
  • Simplifying expressions by combining like terms and using distributive law.
  • Application problems: money, ages, perimeter/area, consecutive integers, averages, and simple mixtures.

Tips

  • Group like terms (same variables and same exponents) to simplify.
  • Use the distributive law a(b + c) = ab + ac to remove brackets.
  • Keep track of units (rupees, cm, years) so the answer is meaningful.
📌 Examples
  • Example 1 — Form an expression: 'Seven more than thrice a number' → 3x + 7. If x = 4, evaluate: 3(4) + 7 = 12 + 7 = 19.
  • Example 2 — Simplify: 5x + 3 + 2x − 8 = (5x + 2x) + (3 − 8) = 7x − 5.
  • Example 3 — Distributive law: 4(x + 6) = 4x + 24. If x = 2, value = 8 + 24 = 32.
  • Example 4 — Age problem: 'Asha is 5 years older than Ravi. If Ravi is x years old, write Asha's age.' → x + 5. If Ravi is 12, Asha is 17.
  • Example 5 — Perimeter application: A rectangle has length (x + 3) cm and width (x − 2) cm. Perimeter = 2[(x + 3) + (x − 2)] = 2(2x + 1) = 4x + 2 cm.
  • Example 6 — Money problem: 'Riya has ₹50 and saves ₹y every week. Amount after 6 weeks?' → 50 + 6y. If y = 30, amount = 50 + 180 = ₹230.
🧮 Formulas
  1. Distributive property: a(b + c) = ab + ac
  2. Combine like terms: ax + bx = (a + b)x
  3. Multiply monomials: (ax^m)(bx^n) = ab x^(m+n)
  4. Value of expression: substitute variable(s) with numbers and compute
  5. Perimeter of rectangle: 2(l + w) → if l = (x + a), w = (x + b) then P = 2(2x + a + b)
  6. Area of rectangle: l × w → (x + a)(x + b) = x^2 + (a + b)x + ab (useful for simple expansions)
📊 Visual ideas
Plot table of values for a linear expression y = 2x + 3 (choose x = −3, −2, −1, 0, 1, 2, 3). Draw the line through the points to visualize how y changes with x.
Use a balance-scale sketch to show an equation like x + 5 = 12. Visualize removing 5 from both sides to keep balance and find x = 7.
Draw bar models for word problems: e.g., represent 'A has 3 times as many apples as B' with one bar split into 3 equal parts (3x) and another into 1 part (x).
Area model (rectangular) to visualize multiplication of binomials: show (x + 2) by (x + 3) as a rectangle partitioned into x×x, x×3, 2×x, and 2×3 to get x^2 + 5x + 6.

Key Concepts

Variable
A symbol (usually a letter) that represents an unknown or changing quantity.
Constant
A fixed numerical value that does not change.
Coefficient
The numerical factor multiplied by a variable in a term.
Term
A single number, variable, or product of numbers and variables separated by plus or minus signs.
Monomial
An algebraic expression consisting of a single term.
Binomial
An algebraic expression made of exactly two terms added or subtracted.
Trinomial
An algebraic expression consisting of three terms.
Polynomial
A sum or difference of one or more monomials (terms with nonnegative integer exponents).
Degree of a term
The sum of the exponents of the variables in that term.
Degree of a polynomial
The highest degree among all its nonzero terms.
Like terms
Terms having the same variables raised to the same powers (may have different coefficients).
Unlike terms
Terms that are not like terms (different variables or different powers).
Algebraic expression
A combination of numbers, variables and mathematical operations (no equals sign).
Linear expression
A polynomial of degree 1 (variable appears to the first power only).
Constant term
A term in an expression or polynomial that contains no variables.
Factor
A quantity that multiplies with others to give a product.
Factorization
Writing an expression as a product of its factors.
Algebraic identity
An equality involving algebraic expressions that holds for all permissible values of the variables.
Simplification
The process of rewriting an expression in a simpler or more compact form by combining like terms and removing parentheses.
Substitution
Replacing a variable with a specific number to evaluate an expression.

End-of-Chapter Trial Paper & Test Questions

Topic-wise questions to test your understanding of every concept in this chapter.

  1. Which of the following is a binomial? / निम्नलिखित में से कौन-सा द्विपद है? (a) 3x (b) 2x + 5y (c) x² + x + 1 (d) 7
    Show answer

    (b) 2x + 5y / (b) 2x + 5y — A binomial has exactly two terms; 2x + 5y has two unlike terms separated by +. / द्विपद में ठीक दो पद होते हैं; 2x + 5y में दो असमान पद हैं।

  2. What is the coefficient of x in the expression 7x² − 5x + 3? / व्यंजक 7x² − 5x + 3 में x का गुणांक क्या है? (a) 7 (b) 3 (c) −5 (d) 5
    Show answer

    (c) −5 / (c) −5 — The term containing x (first power) is −5x, so the coefficient is −5. / x (पहली घात) वाला पद −5x है, इसलिए गुणांक −5 है।

  3. Simplify: 4x + 3y − 2x + y = ? / सरल करें: 4x + 3y − 2x + y = ? (a) 2x + 4y (b) 6x + 4y (c) 2x − 4y (d) 6x − 4y
    Show answer

    (a) 2x + 4y / (a) 2x + 4y — Combine like terms: (4x − 2x) + (3y + y) = 2x + 4y. / समान पदों को मिलाएं: (4x − 2x) + (3y + y) = 2x + 4y।

  4. The degree of the polynomial 5x³ − 2x + 9 is ______. / बहुपद 5x³ − 2x + 9 की घात ______ है।
    Show answer

    3 / 3 — The highest power of the variable x in the polynomial is 3 (in the term 5x³). / बहुपद में चर x की सबसे बड़ी घात 3 है (पद 5x³ में)।

  5. In the expression 8a²b − 3ab + 7, the terms 8a²b and −3ab are ______ terms. / व्यंजक 8a²b − 3ab + 7 में पद 8a²b और −3ab ______ पद हैं।
    Show answer

    unlike (असमान) — They have the same variables a and b but with different powers (a²b vs ab), so they are unlike terms. / उनमें एक ही चर a और b हैं लेकिन घातें अलग हैं (a²b बनाम ab), इसलिए ये असमान पद हैं।

  6. True or False: 3x + 4y can be simplified to 7xy by combining like terms. / सत्य या असत्य: 3x + 4y को समान पदों के मेल से 7xy में सरल किया जा सकता है।
    Show answer

    False / असत्य — 3x and 4y are unlike terms (different variables), so they cannot be combined. / 3x और 4y असमान पद हैं (भिन्न चर), इसलिए इन्हें मिलाया नहीं जा सकता।

  7. Evaluate 2x² − 3x + 5 when x = 2. Show your working. / x = 2 रखने पर 2x² − 3x + 5 का मान ज्ञात कीजिए। हल सहित।
    Show answer

    2(4) − 3(2) + 5 = 8 − 6 + 5 = 7 / 2(4) − 3(2) + 5 = 8 − 6 + 5 = 7 — Substitute x = 2, compute powers first (BODMAS), then multiply, then add/subtract. / x = 2 रखें, पहले घात (BODMAS) फिर गुणा, फिर जोड़-घटाव।

  8. Factorise: 12ab + 18a. / गुणनखंड कीजिए: 12ab + 18a।
    Show answer

    6a(2b + 3) / 6a(2b + 3) — The GCF of 12ab and 18a is 6a; dividing each term gives 2b + 3 inside the bracket. / 12ab और 18a का महत्तम समापवर्तक 6a है; प्रत्येक पद को 6a से भाग देने पर कोष्ठक में 2b + 3 आता है।

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