Chapter 9 — Algebraic Expressions And Identities

What is an algebraic expression? An algebraic expression is a combination of numbers, variables (letters that stand for unknowns) and arithmetic operations (addition, subtraction, multiplication, division, powers). Examples: 3x, 2x + 5, 4y^2 − 7.

Parts of an algebraic expression

• Variable: a letter like x, y that can take values.
• Constant: a fixed number (e.g., 5, −3).
• Coefficient: numerical factor of a term (in 3x, 3 is the coefficient).
• Term: a single part of an expression separated by + or − (e.g., 2x, −7y^2).
• Like terms: terms with the same variable(s) raised to the same power (3x and 5x). Unlike terms cannot be combined (2x and 3y).
• Degree: the highest power of the variable in the term (degree of 5x^3 is 3). Degree of a polynomial is the maximum degree among its terms.

What you can do with expressions

• Evaluate: substitute a number for each variable (e.g., if x = 2, evaluate 3x + 1 = 7).
• Simplify: combine like terms and do arithmetic to write a shorter equivalent expression (e.g., 3x + 5x − 2 = 8x − 2).
• Operate: add, subtract, multiply and (sometimes) divide expressions using algebraic rules.

Expression vs Equation: An expression is a mathematical phrase (no ‘=’). An equation connects two expressions with ‘=’ and states equality (e.g., 2x + 3 = 7).

Why it matters: Algebraic expressions let you model real situations compactly — formulas for area, cost, distance, simple business and science relationships — and prepare you to solve equations and real problems.

An algebraic expression is a combination of numbers, variables (letters) and algebraic operations (addition, subtraction, multiplication, division, powers) but not an equality sign. To classify algebraic expressions we use the number of terms and the highest power (degree) of the variable(s).

• Term, coefficient, variable and constant: A term is a product of numbers and variables (for example 5x^2y). The numerical factor is the coefficient (5). A pure number like 7 is a constant.
• Monomial: An expression with only one term. Examples: 7, 3x, -2ab^2. (Single term; degree of a monomial = sum of exponents of its variables.)
• Binomial: An expression with two unlike terms added or subtracted. Examples: x + 5, 3x^2 - 2x.
• Trinomial: An expression with three unlike terms. Example: x^2 + 5x + 6.
• Polynomial: A sum (or difference) of one or more monomials. Polynomials are classified by the number of terms (monomial, binomial, trinomial) or by degree (linear, quadratic, cubic, etc.). General form for one variable: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where coefficients a_i are constants and a_n ≠ 0.
• Constant polynomial: A polynomial with degree 0 (e.g., 5). The zero polynomial (0) is special — its degree is usually not defined.
• Like and unlike terms: Like terms have exactly the same variable part (same variables raised to the same powers). Only like terms can be added or subtracted directly.
• Degree of a polynomial: For a single-variable polynomial, it is the highest exponent of that variable. For multivariable terms, degree of a term is the sum of exponents of all variables; the polynomial's degree is the maximum such sum among its terms.

Procedure to classify an expression: count the number of unlike terms to decide mono/bi/tri/polynomial; find the highest power (or sum of powers for multivariable) to get the degree; check whether terms are like or unlike to simplify.

Definition: Terms in an algebraic expression are like terms if they have exactly the same variable part (same variables raised to the same powers). Their coefficients may be different. Terms that do not have the same variable part (or same powers) are called unlike terms.

Rules to identify like terms:

• Compare the variable part: variables and their exponents must match exactly. Example: 3x^2y and −5x^2y are like terms.
• Order of factors does not matter: 2xy and 3yx are like terms (because xy = yx).
• Constants (terms with no variables) are like terms with each other: 7 and −2 are like terms.
• Zero behaves as 0·(any variable expression) so 0 can be treated as like any term when simplifying.

How to combine like terms:

1. Identify and group like terms.
2. Add or subtract their coefficients, keeping the common variable part unchanged.
3. Write the result: ax + bx = (a + b)x.

Short worked example: Simplify 4x + 3x^2 − 2x + 7 − 5x^2.

Group like terms: (4x − 2x) + (3x^2 − 5x^2) + 7 = 2x + (−2x^2) + 7 = 2x − 2x^2 + 7.

Important notes: You cannot add unlike terms (for example 5x and 3xy are unlike). To combine terms, the variable parts must match exactly.

What is an algebraic expression? An algebraic expression is a combination of numbers, variables and arithmetic operations (like 3x + 2, 5a - 4b + 7). Terms are parts of an expression separated by + or − signs.

Like terms and unlike terms
Like terms are terms that have exactly the same variable part (same variables raised to the same powers). Example: 5x and −2x are like terms; 3x^2 and x^2 are like terms. Unlike terms have different variable parts (for example, 3x and 4y, or x and x^2).

Basic idea of addition and subtraction
To add or subtract algebraic expressions, you combine only like terms (add or subtract their coefficients). If necessary, first remove parentheses by using the distributive law and change signs for subtraction.

Step-by-step method

1. Remove parentheses: Use distributive property. For subtraction, multiply each term inside the parentheses by −1.
2. Arrange terms so that like terms appear together (group similar variable parts).
3. Add or subtract the coefficients of like terms; keep the common variable part unchanged.
4. Write the result in a simplified standard form (usually descending powers or grouped by variables).

Important remarks
Subtraction can be treated as adding the additive inverse: A − B = A + (−1)B. The commutative and associative laws apply to addition of expressions (you can reorder and regroup like terms).

Multiplication of algebraic expressions means multiplying two or more algebraic terms (monomials, binomials, trinomials, etc.) using the same rules as numerical multiplication together with algebraic laws for variables. Key ideas are:

• Monomial × monomial: multiply numerical coefficients and add exponents of like bases. Example: 2x^2 × 3x^3 = (2×3) x^(2+3) = 6x^5.
• Monomial × polynomial: use the distributive law: a(b + c + ...) = ab + ac + .... Multiply the monomial with each term of the polynomial and then simplify by combining like terms.
• Polynomial × polynomial: multiply each term of the first polynomial by each term of the second (full distribution). For two binomials use FOIL (First, Outer, Inner, Last) as a convenient mnemonic.
• Signs and like terms: apply sign rules (±) when multiplying and always collect like terms (same variable powers) after multiplication.

Special products (identities) are useful shortcuts when multiplying certain expressions. Examples: (a + b)^2, (a − b)^2, (a + b)(a − b).

Step-by-step approach:

1. Identify type of expressions (monomial, binomial, etc.).
2. Apply distributive law: multiply every term of one expression by every term of the other.
3. Simplify: add or subtract like terms and arrange in standard form (descending powers).

Example outline inside HTML: Multiply (x + 2)(x − 3): apply FOIL → x·x + x·(−3) + 2·x + 2·(−3) = x^2 − 3x + 2x − 6 = x^2 − x − 6.

What are standard algebraic identities?
Standard algebraic identities are algebraic equalities that hold for all values of the variables. They help expand, simplify and factor expressions quickly without multiplying each term separately.

Common identities and short derivations

• (a + b)² = a² + 2ab + b²
  Derivation: (a + b)(a + b) = a(a + b) + b(a + b) = a² + ab + ab + b² = a² + 2ab + b².
• (a - b)² = a² - 2ab + b²
  Derivation: (a - b)(a - b) = a² - ab - ab + b² = a² - 2ab + b².
• a² - b² = (a - b)(a + b)
  Derivation: Multiply the right side: (a - b)(a + b) = a² + ab - ab - b² = a² - b².
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
  Derivation: Square (a + b + c) and collect like terms.

How to use them
Use expansion identities to expand expressions without multiplying term by term. Use the difference of squares to factor expressions or compute products of numbers close to each other quickly.

Quick tips

• To square numbers near a base: 99² = (100 - 1)² = 100² - 2·100·1 + 1 = 9801.
• To multiply numbers like 52 × 48: write as (50 + 2)(50 - 2) = 50² - 2² = 2500 - 4 = 2496 (difference of squares).

These identities are tools for simplification, factorization and mental arithmetic and form a foundation for algebraic manipulations in higher classes.

Factorisation is the process of writing an algebraic expression as a product of two or more simpler expressions (factors). It is the reverse of expansion. Factoring helps simplify expressions, solve equations, and find zeros of polynomials.

Common methods of factorisation:

1. Factor out the Greatest Common Factor (GCF): Identify the largest expression that divides every term and factor it out. Example: 12x + 8y = 4(3x + 2y).
2. Factor by grouping: Group terms that have a common factor and factor each group, then factor the common binomial. Example: x^3 + x^2 + x + 1 = x^2(x + 1) + 1(x + 1) = (x + 1)(x^2 + 1).
3. Factor quadratic trinomials (simple case ax^2 + bx + c): Find two numbers whose product = a·c and sum = b; split the middle term and factor by grouping. Example: x^2 + 5x + 6 = (x + 2)(x + 3).
4. Use special identities (recognise patterns and use corresponding factor forms):
   • Perfect square: a^2 + 2ab + b^2 = (a + b)^2
   • Perfect square: a^2 - 2ab + b^2 = (a - b)^2
   • Difference of squares: a^2 - b^2 = (a - b)(a + b)

General steps to factor an expression:

1. Look for a common factor and factor it out.
2. If still not simplest, check for grouping or special identities.
3. For quadratics, try splitting the middle term or use product-sum method.
4. Verify by multiplying the factors (expanding) to get the original expression.

Why factorisation matters: Factorisation reveals roots/zeroes of polynomials, simplifies algebraic manipulation in equations and formulas, and models composite quantities in real life (areas, costs, grouping problems).

What is an algebraic expression? An algebraic expression is a combination of numbers, variables (like x, a, b) and arithmetic operations (addition, subtraction, multiplication, division, powers). Examples: 3x + 5, 2x^2 - x + 7.

Value of an algebraic expression (Evaluation)
To find the value of an algebraic expression for given values of its variables:

• Substitute the given number(s) for the variable(s).
• Follow the order of operations: brackets → powers → multiplication/division → addition/subtraction.
• Simplify by combining like terms and performing arithmetic.

Key points when evaluating:

• A coefficient is the numerical factor of a term (in 4x, 4 is the coefficient).
• Like terms have the same variable part (2x and -5x are like terms).
• Be careful with signs and brackets — use parentheses when substituting negative values: e.g., (−2)^2 = 4 but −2^2 = −4.

What is an identity? An identity is an equality that is true for all permissible values of the variables. Example: (a + b)^2 = a^2 + 2ab + b^2 is an identity because both sides are equal for every a and b.

Verification of identities — two common methods:

1. Algebraic verification (symbolic): Expand or simplify one or both sides using algebraic laws (distributive, associative, commutative) until both sides are identical. If they simplify to the same expression, the identity is verified.
2. Numerical verification (substitution): Substitute several different numerical values for the variables. If both sides give equal results for many arbitrary values (not just one), this supports the identity. (Note: numerical checks are supportive but not a formal proof; algebraic simplification is the formal proof.)

Common algebraic laws used:

• Distributive law: a(b + c) = ab + ac
• Commutative: a + b = b + a, ab = ba
• Associative: (a + b) + c = a + (b + c)
• Power rules for small integer exponents used in expansions (square, cube formulas)

Tips and common mistakes:

• Always use parentheses when substituting negative numbers: x = −3 into x^2 gives (−3)^2 = 9.
• Do not cancel unlike terms — only like terms combine.
• For identities involving two variables, algebraic simplification is the definitive proof; substitution should use several values to build confidence.

What this topic means
"Applications and Problem Solving" uses algebraic expressions and basic identities to model real-life situations, simplify calculations and solve word problems. The main skills are: translating words into expressions/equations, using identities to simplify, solving for unknowns, and interpreting answers.

Steps to solve problems

1. Read carefully: identify what is asked and the quantities involved.
2. Choose variables: assign a letter (usually x) to the unknown quantity.
3. Form expressions/equations: translate relationships and conditions into algebraic expressions using operations and identities.
4. Simplify: use algebraic identities (like (a+b)^2 = a^2 + 2ab + b^2) to reduce complexity.
5. Solve: perform algebraic steps to find the variable(s).
6. Check and interpret: substitute back into the context to ensure the solution makes sense (no negative ages, etc.).

Why identities help
Identities let you expand or factor expressions quickly so you can compare, combine or solve equations. For example, using a^2 - b^2 = (a - b)(a + b) can turn a subtraction of squares into a product that is easy to factor or evaluate.

Types of real-life problems
Common applications for Class 8 include: ages, money, consecutive numbers, dimensions and area/perimeter problems, mixtures, simple motion or rate problems where relationships are linear, and simple algebraic simplifications needed in arithmetic shortcuts.