C2.2 Algebraic manipulation

Learning objectives

C2.2.1Simplify expressions by collecting like terms.• Simplify expressions by collecting like terms.C2.2.2Expand products of algebraic expressions.• Expand products of algebraic expressions.C2.2.3Factorise by extracting common factors• Factorise by extracting common factors. Simplify means give the answer in its simplest form, e.g. 2a + 3b + 5a – 9b = 7a – 6b. e.g. expand 3x(2x – 4y). Includes products of two brackets involving one variable, e.g. expand (2x + 1)(x – 4). Factorise means factorise fully, e.g. 9x2 + 15xy = 3x(3x + 5y).

Like terms have exactly the same variable part, including the same powers. They represent the same kind of quantity, so only their numerical coefficients can be combined.

Pair	Like terms?	Reason
$3a$ and $-5a$	yes	both have variable part $a$
$4y^2$ and $-y^2$	yes	both have variable part $y^2$
$2a$ and $2b$	no	different variables
$x$ and $x^2$	no	different powers

Group each set of like terms, add or subtract their coefficients, and keep the common variable part unchanged. For example, $2a+3b+5a-9b=(2+5)a+(3-9)b=7a-6b$ .

Treat different powers as different groups. Thus $4y^2+3y-y^2+2y=(4-1)y^2+(3+2)y=3y^2+5y$ . Neither $y^2$ nor $y$ can be absorbed into the other.

A sign belongs to the term immediately after it. In $a-2b-3a+7b$ , collect $a-3a=-2a$ and $-2b+7b=5b$ , giving $-2a+5b$ .

Do not combine unlike terms by adding their coefficients: $3a+4b$ is already simplified. Collecting terms changes coefficients, not variables or powers; for instance, $3x+7x-4x=6x$ , not $6x^2$ .

Expansion uses the distributive law to replace a product containing brackets with an equivalent sum or difference. Every term in one factor must multiply every term in the other factor.

Form	Products to write	Then simplify
$k(a+b)$	$ka+kb$	collect if possible
$k(a-b)$	$ka-kb$	keep the second sign
$(a+b)(c+d)$	$ac+ad+bc+bd$	collect like middle terms

Multiply the outside term by each term inside. The syllabus example becomes $3x(2x-4y)=3x\cdot2x+3x\cdot(-4y)=6x^2-12xy$ . The variable products must also be multiplied.

For $(2x+1)(x-4)$ , write all four products: $2x^2-8x+x-4$ . The two middle terms are like terms, so the simplified result is $2x^2-7x-4$ .

A minus before a bracket changes the sign of every term in that bracket. For $4(x-5)-(3-2x)$ , expansion gives $4x-20-3+2x=6x-23$ .

Do not stop after multiplying only the first terms, and do not drop a negative product. Expansion rewrites an expression; it does not solve an equation. A quick check is that substituting the same value of $x$ before and after expansion gives the same result.

Factorising by a common factor reverses expansion: write an expression as a product by taking outside the greatest factor shared by every term.

Part of the common factor	How to choose it	Example
number	greatest common divisor of all coefficients	for 9 and 15, choose 3
variable	include only variables present in every term	$x$ is common to $x^2$ and $xy$
power	use the smallest shared power	$a^2$ and $ab$ share $a$

Find the greatest common factor, divide every term by it to form the bracket, then multiply back to check. For $9x^2+15xy$ , the greatest common factor is $3x$ , so $9x^2+15xy=3x(3x+5y)$ .

Factorisation is complete only when the bracket has no further common factor. For $18x^2-12x$ , $2x(9x-6)$ is only partial; extracting $6x$ gives the fully factorised form $6x(3x-2)$ .

Preserve the operation signs when dividing each term. For $5x-20x^2$ , extracting $5x$ gives $5x(1-4x)$ because $5x\div5x=1$ and $-20x^2\div5x=-4x$ .

Only extract a factor shared by every term. Do not use quadratic-factor methods here: this Core objective is common-factor extraction. Expansion of the final product must reproduce the original expression exactly.

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—C2.4 Indices IIRead

C2.2 Algebraic manipulation

Learning objectives

Collect like terms without changing their variable part

Expand every product across algebraic brackets

Factorise fully by extracting the greatest common factor

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