Chapter 14 — Factorisation

What is factorisation?
Factorisation is the process of writing a number or an algebraic expression as a product of two or more factors. For algebraic expressions it means reversing multiplication (or the distributive law) to express the expression as a product of simpler expressions.

Why learn factorisation?
It helps simplify expressions, solve equations, find zeros of polynomials, and model real-life grouping problems. Factorisation is an essential tool for solving quadratic equations and simplifying algebraic fractions.

Basic idea (using distributive law):
If ab + ac = a(b + c), then factoring means taking the common factor out: ab + ac = a(b + c). Factorisation is simply applying this idea in various forms.

Common methods of factorisation

• 1. Taking common factor: Identify and factor out the greatest common factor (GCF). Example: 12x + 8 = 4(3x + 2).
• 2. Factorisation by grouping: Group terms in pairs (or groups) and factor common factors from each group, then factor the resulting common binomial. Example: ax + ay + bx + by = (a + b)(x + y).
• 3. Difference of squares: a^2 - b^2 = (a - b)(a + b). Example: x^2 - 9 = (x - 3)(x + 3).
• 4. Perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2. Example: x^2 + 6x + 9 = (x + 3)^2.
• 5. Simple quadratic factorisation: For ax^2 + bx + c (often a=1 in Class 8), find two numbers whose product is c and sum is b, then split and factor by grouping. Example: x^2 + 5x + 6 = (x + 2)(x + 3).

Steps / Tips

1. Always look for a common factor first.
2. If no single common factor, try grouping terms.
3. Recognise special patterns (difference of squares, perfect squares).
4. Check by expanding (multiplying) the factors to get the original expression.

Checking result
Multiply the factors you obtained (use distributive law) — you should get the original expression.

Summary: Factorisation converts sums into products: it simplifies expressions and reveals roots/zeros when factors equal zero. Mastery comes from recognizing patterns and practising different methods.

What is a polynomial? A polynomial in one variable x is an expression made by adding or subtracting terms of the form a x^n, where a is a constant (coefficient) and n is a non‑negative integer (0, 1, 2, ...). Examples: 3x^2 + 5x − 4, 7, x^3 − 2x + 1.

Terms, coefficients and constant term

• Each addend like 3x^2, 5x or −4 is called a term.
• The number multiplying the power of x (for 3x^2 it is 3) is the coefficient.
• The term with x^0 (i.e. no x) is the constant term (for 3x^2 + 5x − 4 the constant is −4).

Degree of a term

• The degree of a term a x^n is the exponent n. Example: degree of 5x^3 is 3; degree of 7x is 1; degree of 9 is 0.

Degree of a polynomial

• The degree of a polynomial is the highest degree (largest exponent) among its nonzero terms. Example: degree of 4x^5 − x^2 + 7 is 5.
• Special cases: the degree of a nonzero constant polynomial (like 7) is 0. The zero polynomial (0) has no well‑defined degree (some advanced texts use −∞ as a convention), so for Class 8 say: degree is not defined.

Classification by number of terms

• Monomial — one term (e.g., 6x^2).
• Binomial — two terms (e.g., x + 5).
• Trinomial — three terms (e.g., x^2 − 3x + 2).

Important observations

• Only nonnegative integer exponents are allowed in polynomials. Expressions like x^(−1) or x^(1/2) are not polynomials.
• When adding like terms (terms with same power of x), combine their coefficients before deciding degree.

Why degree matters

• The degree tells us the general shape and behavior of the polynomial for large |x|. The term with the highest degree is called the leading term and its coefficient is the leading coefficient.

What it means
Factorisation by taking common factor is the process of rewriting a sum (or difference) of terms as a product, by extracting a factor that is common to all terms. It uses the distributive property: a(b + c) = ab + ac. In reverse, given ab + ac, we write it as a(b + c).

Why and when we use it
We use this method to simplify expressions, solve equations, find zeros of polynomial functions, and make algebraic manipulations easier.

Steps (simple algorithm)

1. Look at each term and identify common numerical factors (find the HCF of the coefficients).
2. Look at the variables in each term and take out every variable with the smallest power common to all terms.
3. Write the common numerical factor and common variables as one factor, then divide each original term by this factor and place the results inside parentheses separated by + or −.
4. Check by expanding (multiplying) to ensure you get the original expression.

Key notes

• If all terms have a negative common factor, you may factor out the negative number so the first term inside the parenthesis is positive (helps in some cases).
• Factoring out the greatest common factor (GCF/HCF) makes the inside parentheses simplest.
• Factoring does not change the value of the expression; it only rewrites it as a product.

Worked pattern (distributive law)
Given terms T1, T2: if T1 = a·u and T2 = a·v then T1 + T2 = a(u + v). Example: 6x + 9 = 3(2x + 3).

What it is: Factorisation by grouping is a method to factor algebraic expressions (usually with 4 terms or a quadratic by splitting the middle term) by grouping terms that have a common factor. After grouping, a common binomial factor is taken out to produce a product of two factors.

When to use: Use it when an expression has four terms, or when a quadratic ax² + bx + c can be written by splitting the middle term (bx) into two terms whose sum is b and whose product is a·c.

General idea (four-term case):

• Start with: ax + ay + bx + by
• Group as: (ax + ay) + (bx + by)
• Factor common terms in each group: a(x + y) + b(x + y)
• Factor the common binomial: (x + y)(a + b)

Steps for a quadratic ax² + bx + c (split-the-middle method):

• Find two numbers m and n such that m + n = b and m·n = a·c.
• Rewrite bx as m x + n x, giving ax² + m x + n x + c.
• Group: (ax² + m x) + (n x + c).
• Factor each group and then take out the common binomial factor to get (px + q)(rx + s).

Why it works: Grouping uses distributive property in reverse. By factoring common parts from grouped terms you reverse the expansion (a+b)(c+d)=ac+ad+bc+bd to get back to two factors.

Check your answer: Multiply the factors to verify you return to the original expression.

What it is: Factorisation using identities means rewriting an expression as a product using well-known algebraic formulas (special products). Instead of expanding, we reverse the expansion to write expressions as factors. This is useful for simplifying expressions and solving equations.

Common identities (forward form):

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• (a + b)(a - b) = a^2 - b^2 (difference of squares)

How to factor using these identities:

• Look for a perfect square trinomial: a^2 ± 2ab + b^2 → factor as (a ± b)^2. Identify a and b by comparing square terms.
• Look for a difference of two squares: a^2 - b^2 → factor as (a - b)(a + b).
• If coefficients are perfect squares (e.g., 9x^2, 16), extract their square roots first (e.g., 9x^2 - 24x + 16 = (3x - 4)^2).

Why it helps: Factorised form reveals roots (zeros) of polynomials, simplifies evaluation (mental math), and connects algebra to geometry (areas).

Geometric idea: The expansion (a + b)^2 corresponds to a square of side (a + b) cut into one a×a square, one b×b square and two a×b rectangles. The identity (a + b)^2 = a^2 + 2ab + b^2 follows directly from this picture.

What is a quadratic trinomial? A quadratic trinomial is a polynomial of the form ax^2 + bx + c where a, b and c are numbers and a ≠ 0. Factorising a quadratic trinomial means writing it as a product of two binomials, typically (px + q)(rx + s), so that when multiplied you get back ax^2 + bx + c.

Why factorise? Factorisation helps solve equations, simplify expressions, find zeros/roots of quadratic functions and model many real situations.

Main idea / condition: If ax^2 + bx + c = (px + q)(rx + s) then:

• p·r = a
• q·s = c
• p·s + q·r = b

Methods:

• Case a = 1 (simple): Find two integers m and n such that m·n = c and m + n = b. Then x^2 + bx + c = (x + m)(x + n).
• General case (split the middle term): For ax^2 + bx + c find two numbers m and n such that m·n = a·c and m + n = b. Rewrite bx as mx + nx, then factor by grouping.
• Factor by grouping: After splitting the middle term, group terms into two pairs and factor a common binomial factor.
• Special trinomials: Perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2.

Tips: First check for a greatest common factor (GCF) in all terms and take it out. If no integer factor pair exists, the trinomial may not factorise into integer-coefficient binomials.

Factorisation is the process of writing an algebraic expression as a product of simpler expressions called factors. For expressions with more than one variable (for example x and y), the same basic methods used for single-variable expressions apply: taking a common factor, factoring by grouping, and recognising special products such as difference of squares and perfect square trinomials.

Key methods:

• Factor out the greatest common factor (GCF): Look for a common numerical factor and common powers of variables. Example pattern: axmyn + bxmyn = (a+b)xmyn.
• Factor by grouping: Group terms into pairs (or groups) that have a common factor, factor each group, and then factor out the common binomial. Useful for four-term expressions and some trinomials.
• Use special product identities: Recognise patterns like a^2 - b^2 = (a-b)(a+b), (a+b)^2 = a^2 + 2ab + b^2, and (a-b)^2 = a^2 - 2ab + b^2. Here a and b can be expressions in x and y.

Step-by-step strategy:

1. Rewrite the expression clearly and look for a common numerical factor and common variable powers.
2. If no single common factor removes all terms, try grouping terms so each group has a common factor.
3. Check whether the expression matches a special product form (difference of squares, perfect square trinomial).
4. After factoring, always expand (multiply the factors) to verify you get the original expression.

Tips:

• Treat each term as a product of number and variables. For example 6x^2y = 6 * x * x * y.
• When factoring trinomials in x and y, consider one variable (say x) as the main variable and factor like a quadratic in x with coefficients involving y.
• Keep signs organized; factoring out a negative sign can simplify the appearance.

Zero-Product Property: If the product of two or more factors is zero, then at least one of the factors must be zero. In symbols, if A·B = 0 then A = 0 or B = 0 (or both).

Why it helps: Many algebraic equations can be turned into a product of simpler expressions (factors). Once we factor an expression and set it equal to zero, we can apply the zero-product property to find the solutions (roots).

Steps to solve an equation by factorisation:

• Write the equation in standard form, usually with 0 on one side (e.g., bring all terms to left).
• Factor the expression completely (take out common factors, use patterns like difference of squares, perfect square trinomials, or split the middle term for quadratics).
• Apply the zero-product property: set each factor equal to zero and solve each simple equation.
• Check solutions in the original equation if necessary.

Common factorisation methods used in Class 8:

• Factor out the greatest common factor (GCF): a·b + a·c = a(b + c).
• Difference of squares: x^2 - y^2 = (x - y)(x + y).
• Perfect square trinomials: x^2 + 2xy + y^2 = (x + y)^2 and x^2 - 2xy + y^2 = (x - y)^2.
• Split the middle term for simple quadratics ax^2 + bx + c when factorable.

Important note: Factorisation and the zero-product property only apply after the expression is written as a product equal to zero. If the expression cannot be factored over integers, other methods (like completing the square or quadratic formula) are used in higher classes.

Factorisation is reversing multiplication: writing an expression as a product of factors. Common mistakes cause wrong factors and wrong answers. This topic shows typical pitfalls and reliable verification methods to ensure factorisation is correct.

• Common pitfalls
  • Signs: forgetting to change signs when taking out a negative common factor or splitting middle term (e.g., -x vs +x).
  • Not factoring completely: stopping after one step when further common factors or identities apply.
  • Incorrect grouping: grouping terms in a way that doesn’t produce common factors.
  • Misapplying identities: using (a+b)^2 instead of a^2+2ab+b^2 or misusing difference of squares.
  • Assuming integer factors always exist for ax^2+bx+c; sometimes factors are not integers.
  • Dropping terms when simplifying or cancelling (especially dividing by zero or an expression that can be zero).
• Reliable verification methods
  1. Multiply the factors (expand) and check you get the original expression (use distributive law).
  2. Substitute one or two simple numerical values for the variable(s) into both the original and the factored form — results must match.
  3. Compare coefficients: expand and equate coefficients of like powers to the original coefficients.
  4. Check zeros/roots: for a quadratic, the factors (x - r1)(x - r2) should give roots r1 and r2; plug these into the original to get zero.
• Quick checklist before finalizing answer
  1. Have you taken out the greatest common factor (GCF)?
  2. Have you applied special identities correctly (square, difference of squares)?
  3. Is the sign pattern consistent after factoring?
  4. Did you expand or substitute to verify?

Using these checks reduces errors and helps build confidence in factorisation steps.

What this topic covers: Factorisation is rewriting an algebraic expression as a product of simpler expressions (factors). In Class 8 you use factorisation to simplify expressions, solve simple equations, model area/perimeter problems, and make calculations easier.

Why factorisation is useful (applications):

• Solving equations: If P(x) = 0 and P(x) is factorised as (A)(B) = 0, then A = 0 or B = 0 gives solutions quickly.
• Geometry & measurement: Areas and perimeters often give algebraic expressions that can be interpreted as products (e.g., area = length × breadth). Factorisation helps find possible dimensions.
• Number problems & grouping: Splitting items into groups, finding common factors or arranging tiles uses factor ideas.
• Algebraic simplification: Cancelling common factors in fractions or simplifying expressions before evaluation.

Common methods you will use:

• Take out the greatest common factor (GCF).
• Factor by grouping (for 4-term expressions).
• Factor a trinomial x^2 + bx + c by finding two numbers whose product is c and sum is b.
• Use standard identities: difference of squares and perfect square trinomials.

How to use factorisation to solve problems (step-by-step):

1. Translate the word problem into an algebraic expression or equation.
2. Simplify the expression and try to factor it using one of the methods above.
3. If solving an equation, set each factor equal to zero to find variable values.
4. Interpret the solution in the context of the problem (discard negatives if dimensions must be positive, etc.).

Practice problems (with brief approaches):

1. Factor and solve x^2 + 5x + 6 = 0. (Factor to (x+2)(x+3)=0 → x=-2 or x=-3.)
2. The area of a rectangle is given by x^2 + 5x. Find possible dimensions. (Factor x(x+5) → dimensions could be x and x+5.)
3. Factor 4x^2 - 9. (Recognise difference of squares: (2x-3)(2x+3).)
4. Simplify and factor 3x^2 + 6x. (Take GCF 3x: 3x(x+2).)
5. A school buys notebooks at rate (2x + 3) rupees per dozen and total cost is 6x^2 + 9x. Show how many dozens were bought. (Factor total as 3x(2x+3) → dozens = 3x.)

Tips for checking & choosing correct factors:

• Multiply your factors to check you get the original expression.
• Watch signs: for x^2 + bx + c, if c is positive the two numbers have same sign; if c is negative they have opposite signs.
• When factoring ax^2 + bx + c with a ≠ 1, either use splitting the middle term or look for two numbers whose product is a·c and whose sum is b.

Sample small practice set (try before checking solutions):

1. Factor: 5x^2 - 20x
2. Factor: x^2 - x - 12
3. Factor: 6x^2 + 11x + 3
4. Area problem: Area = x^2 + 7x + 10. Find possible whole-number dimensions.
5. Solve: x^2 - 4x = 0 (using factorisation)

How teachers expect answers in Class 8: Show each step: identify common factor or identity used, write the factorised form, and if solving, list all roots and explain which are acceptable in context (e.g., negative roots discarded for lengths).