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Algebra - Expansion and Factorization

Grade 12IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Distributive Law: Multiplying a single term over a bracket or expanding two or more brackets.

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Factorization: The process of writing an expression as a product of its factors (the reverse of expansion).

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Highest Common Factor (HCF): Identifying and extracting the largest shared factor from all terms in an expression.

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Difference of Two Squares (DOTS): A specific pattern where a2−b2a^2 - b^2 is factorized into (a−b)(a+b)(a-b)(a+b).

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Quadratic Trinomials: Factorizing expressions of the form ax2+bx+cax^2 + bx + c by finding factors that multiply to acac and add to bb.

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Factorization by Grouping: Used for expressions with four terms by pairing them to find common binomial factors.

📐Formulae

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

(a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

💡Examples

Problem 1:

Expand and simplify (2x−3)(x+4)(x−1)(2x - 3)(x + 4)(x - 1).

Solution:

(2x2+8x−3x−12)(x−1)=(2x2+5x−12)(x−1)=2x2(x−1)+5x(x−1)−12(x−1)=2x3−2x2+5x2−5x−12x+12=2x3+3x2−17x+12(2x^2 + 8x - 3x - 12)(x - 1) = (2x^2 + 5x - 12)(x - 1) = 2x^2(x-1) + 5x(x-1) - 12(x-1) = 2x^3 - 2x^2 + 5x^2 - 5x - 12x + 12 = 2x^3 + 3x^2 - 17x + 12

Explanation:

Expand the first two brackets using FOIL, simplify the resulting quadratic, and then multiply each term of the quadratic by each term in the third bracket.

Problem 2:

Factorize completely: 16x4−8116x^4 - 81.

Solution:

(4x2)2−92=(4x2−9)(4x2+9)=(2x−3)(2x+3)(4x2+9)(4x^2)^2 - 9^2 = (4x^2 - 9)(4x^2 + 9) = (2x - 3)(2x + 3)(4x^2 + 9)

Explanation:

First, apply the Difference of Two Squares identity. Then, notice that (4x2−9)(4x^2 - 9) is also a difference of two squares and can be factorized further. (4x2+9)(4x^2 + 9) cannot be factorized over real numbers.

Problem 3:

Factorize the trinomial: 6x2−7x−36x^2 - 7x - 3.

Solution:

6x2−9x+2x−3=3x(2x−3)+1(2x−3)=(3x+1)(2x−3)6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (3x + 1)(2x - 3)

Explanation:

We look for two numbers that multiply to 6×−3=−186 \times -3 = -18 and add to −7-7. These numbers are −9-9 and 22. We split the middle term and factorize by grouping.

Problem 4:

Factorize by grouping: ax+ay−2bx−2byax + ay - 2bx - 2by.

Solution:

a(x+y)−2b(x+y)=(a−2b)(x+y)a(x + y) - 2b(x + y) = (a - 2b)(x + y)

Explanation:

Group the first two terms and the last two terms. Extract the common factor aa from the first group and −2b-2b from the second group. Then extract the common binomial (x+y)(x + y).