Manipulating and factorising polynomialsEdexcel International A Level Maths: Revision notes

Section 1

Polynomials and notation

A polynomial is a sum of terms of the form $ax^n$ with $n$ a non-negative integer. Its degree is the highest power of $x$ , e.g. $x^3+4x^2+3x$ has degree $3$ . The notation $f(x)$ names the polynomial and $f(a)$ means 'substitute $x=a$ '. For $f(x)=x^3+4x^2+3x$ , $f(-2)=-8+16-6=2$ . Take care with negative numbers: use brackets, $(-2)^3=-8$ .

Key termspolynomialdegreecoefficient

Common mistake

Writing $(-2)^2$ as $-4$ . Use brackets when substituting negative values.

Section 2

Expanding brackets and collecting like terms

Multiply every term in one bracket by every term in the next, then collect like terms (same power of $x$ ). Example: $(2x-1)(x+3)^2$ . First $(x+3)^2=x^2+6x+9$ . Then $(2x-1)(x^2+6x+9)=2x^3+12x^2+18x-x^2-6x-9=2x^3+11x^2+12x-9$ . A cube such as $(x-2)^3$ is done in two steps: $(x-2)^2=x^2-4x+4$ , then multiply by $(x-2)$ to get $x^3-6x^2+12x-8$ .

Key termslike terms

Common mistake

Squaring a bracket as $(x+3)^2=x^2+9$ . Always write it out as $(x+3)(x+3)$ .

Exam tip

Check an expansion by substituting $x=1$ in the original and the expanded form.

Section 3

Factorising: common factors and quadratics

Always look for a common factor first. For $x^3+4x^2+3x$ , take out $x$ : $x(x^2+4x+3)$ . Then factorise the quadratic: $x(x+1)(x+3)$ . A quadratic $x^2+bx+c$ factorises as $(x+p)(x+q)$ with $p+q=b$ and $pq=c$ . For $ax^2+bx+c$ , find two numbers with product $ac$ and sum $b$ , then split the middle term. A factorised polynomial must be fully factorised: $x(x^2+4x+3)$ is not finished.

Key termscommon factorfully factorised

Exam tip

After factorising, expand your answer mentally to check.

Section 4

Difference of two squares and cubics

The difference of two squares: $a^2-b^2=(a-b)(a+b)$ . So $x^2-9=(x-3)(x+3)$ , and $x^3-9x=x(x^2-9)=x(x-3)(x+3)$ . For a cubic with no constant term, such as $2x^3-8x^2+6x$ , take out the common factor $2x$ to leave a quadratic: $2x(x^2-4x+3)=2x(x-1)(x-3)$ . Polynomials of degree up to $3$ at this level factorise by common factor, then a quadratic or the difference of two squares.

Key termsdifference of two squares

Common mistake

Trying to factorise a sum of two squares such as $x^2+9$ : it does not factorise over the real numbers.

Section 5

Using polynomials in context

In context problems, form an expression for a volume or area, expand and simplify, then factorise to reveal structure. A box with square base side $(x+2)$ and height $(2x-1)$ has volume $(x+2)^2(2x-1)=2x^3+7x^2+4x-4$ . Total surface area $=2(x+2)^2+4(x+2)(2x-1)=10x^2+20x=10x(x+2)$ . Subtracting one polynomial from another: remove the brackets carefully, changing every sign inside the bracket being subtracted.

Common mistake

Subtracting a bracket and changing only the first sign: $-(x^3+3x^2-4)$ changes all three terms.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Manipulating and factorising polynomials

The polynomial $f(x)=x^3+4x^2+3x$ is defined for real $x$ .
Solve $f(x)=0$ .2 marks
The function $g(x)=(2x-1)(x+3)^2$ is defined for real $x$ .
Show that $g(x)=2x^3+11x^2+12x-9$ .2 marks
The polynomials $h(x)=x^3-9x$ and $k(x)=x^2-x-12$ are defined for real $x$ .
Factorise $h(x)$ completely.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).