E2.2 Algebraic manipulation

Learning objectives

E2.2.1Simplify expressions by collecting like terms.• Simplify expressions by collecting like terms.E2.2.2Expand products of algebraic expressions.• Expand products of algebraic expressions.E2.2.3Factorise by extracting common factors.• Factorise by extracting common factors.E2.2.4Factorise expressions of the form: • ax + bx + kay + kby • a2x2 − b2y2• Factorise expressions of the form: • ax + bx + kay + kby • a2x2 − b2y2 • a2 + 2ab + b2 • ax2 + bx + c • ax3 + bx2 + cx.E2.2.5Complete the square for expressions in the form ax2 + bx + c• Complete the square for expressions in the form ax2 + bx + c. Simplify means give the answer in its simplest form, e.g. 2a2 + 3ab – 1 + 5a2 – 9ab + 4 = 7a2 – 6ab + 3. e.g. expand 3x(2x – 4y), (3x + y)(x – 4y). Includes products of more than two brackets, e.g. expand (x – 2)(x + 3)(2x + 1). Factorise means factorise fully, e.g. 9x2 + 15xy = 3x(3x + 5y).

Like terms have exactly the same variable part, including the same powers. Their coefficients can be added or subtracted because they count the same kind of algebraic quantity.

Terms	Like?	Reason
$3x$ and $-5x$	yes	both have variable part $x$
$2a^2$ and $7a^2$	yes	both have variable part $a^2$
$4x$ and $4x^2$	no	the powers differ
$3ab$ and $-2ba$	yes	$ab=ba$

Group like terms, then combine only their coefficients: $4a^2-3ab+2+5a^2+7ab-6=(4+5)a^2+(-3+7)ab+(2-6)=9a^2+4ab-4$ .

A reliable check is to identify the variable part of every term before calculating. Constants are like terms with one another, but a constant is not like a term containing a variable.

Do not combine unlike terms: $3x+2x^2$ cannot become $5x^3$ or $5x^2$ . Simplifying changes the form of an expression, not its value.

Expanding removes brackets by multiplying every term in one factor by every term in the other factor. This is the distributive law: $a(b+c)=ab+ac$ .

For one bracket, multiply the outside term by each inside term: $3x(2x-4y)=6x^2-12xy$ . Keep the sign attached to each term.

$(3x+y)(x-4y)$

Make all four products: $3x^2-12xy+xy-4y^2$ . Then collect like terms to obtain $3x^2-11xy-4y^2$ .

With more than two brackets, expand two factors first, simplify, then multiply by the next factor. For example, start $(x-2)(x+3)(2x+1)$ by finding $(x-2)(x+3)=x^2+x-6$ .

A negative term changes the sign of its product. Missing just one cross-product makes a double-bracket expansion incomplete; multiplying only the first terms is not enough.

Factorising is the reverse of expanding. To factorise fully by extraction, place the greatest factor shared by every term outside one pair of brackets.

Find the greatest common numerical factor. For each variable, take the lowest power present in every term. Divide every original term by that common factor to form the bracket.

$18x^2y+24xy^2$

The greatest common numerical factor is $6$ ; both terms also contain $xy$ . Therefore $18x^2y+24xy^2=6xy(3x+4y)$ .

Expand the result to check: $6xy(3x+4y)=18x^2y+24xy^2$ . If the bracket still has a common factor, the expression has not been factorised fully.

A factor must divide every term. Do not extract a higher variable power than the smallest power common to all terms.

Factorise completely by extracting any common factor first, then recognise the structure that remains. Expanding the final factors should reproduce the original expression.

Structure	Factorised form	Recognition cue
$ax+bx+kay+kby$	$(a+b)(x+ky)$	group pairs with a repeated bracket
$a^2x^2-b^2y^2$	$(ax-by)(ax+by)$	difference of two squares
$a^2+2ab+b^2$	$(a+b)^2$	square ends and twice their product
$ax^2+bx+c$	two linear factors	product gives $ac$ and sum gives $b$
$ax^3+bx^2+cx$	$x(ax^2+bx+c)$ first	every term contains $x$

For $6x^2+7x-20$ , use numbers with product $6(-20)=-120$ and sum $7$ : $15$ and $-8$ . Split and group: $6x^2+15x-8x-20=3x(2x+5)-4(2x+5)=(3x-4)(2x+5)$ .

For $x^3-25x$ , extract $x$ first, then use a difference of squares: $x(x^2-25)=x(x-5)(x+5)$ .

A single expression may need more than one step. For example, $20x^2-45y^2=5(4x^2-9y^2)=5(2x-3y)(2x+3y)$ .

Pattern conditions matter: $a^2+b^2$ is not a difference of squares, and the middle term of a perfect square must be exactly $2ab$ . This card factorises expressions; it does not solve equations.

Completing the square rewrites $ax^2+bx+c$ as a multiple of one squared binomial plus a constant. It reverses the identity $(x+p)^2=x^2+2px+p^2$ .

$x^2+bx+c=\left(x+ rac b2 ight)^2+c-\left( rac b2 ight)^2$

When $a
e1 $, first factor$ a $from the$ x^2 $and$ x $terms. Equivalently,$ ax^2+bx+c=a\left(x+rac{b}{2a}
ight)^2+c-rac{b^2}{4a} $for$ a
e0$.

$2x^2+12x+5=2(x^2+6x)+5=2[(x+3)^2-9]+5=2(x+3)^2-13$ . Half the coefficient of $x$ inside the bracket to obtain $3$ .

Expand to verify: $2(x+3)^2-13=2(x^2+6x+9)-13=2x^2+12x+5$ .

Factor out only from the terms that contain $x$ before completing the square; do not accidentally divide the outside constant. Rewriting the expression is not the same as solving a quadratic equation.

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E2.2 Algebraic manipulation

Learning objectives

Collect like terms to simplify expressions

Expand products using the distributive law

Factorise by extracting the greatest common factor

Choose and apply complete factorisation patterns

Complete the square for a quadratic expression

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