Polynomials and factor theorem Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Polynomials and factor theorem questions. See exactly how to solve problems on expanding, double brackets, polynomial product, perfect square.

expandingdouble bracketspolynomial productperfect squarefactorisingquadratic

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Expand and simplify

(x+3)(x+5)

Worked solution

Write out the product
$(x+3)(x+5)$
We must multiply everything in the first bracket by everything in the second bracket. This is the same double-bracket skill you met with quadratics.
Multiply each pair of terms
$x\cdot x + x\cdot 5 + 3\cdot x + 3\cdot 5$
Take each term in the first bracket and multiply it by each term in the second (often called FOIL). This gives four separate products.
Simplify each product
$x^2 + 5x + 3x + 15$
Work out each multiplication. Remember x times x is x squared.
Collect like terms
$x^2 + 8x + 15$
The two x-terms, 5x and 3x, add to give 8x. Nothing else can be combined.

Answer

x^2 + 8x + 15

Question 2

2 markseasy

Expand and simplify

(2x-1)(x+4)

Worked solution

Write out the product
$(2x-1)(x+4)$
Set up the two brackets ready to multiply. Watch the negative sign in front of the 1.
Multiply each pair of terms
$2x\cdot x + 2x\cdot 4 + (-1)\cdot x + (-1)\cdot 4$
Multiply every term in the first bracket by every term in the second. Keep the minus sign attached to the 1.
Simplify each product
$2x^2 + 8x - x - 4$
Carefully evaluate each multiplication, keeping track of signs.
Collect like terms
$2x^2 + 7x - 4$
The x-terms 8x and -x combine to 7x. The answer is now fully simplified.

Answer

2x^2 + 7x - 4

Question 3

3 markseasy

Expand and simplify

(x+2)(x^2+3x+1)

Worked solution

Split the first bracket
$x(x^2+3x+1) + 2(x^2+3x+1)$
Multiply the whole quadratic by x, then multiply the whole quadratic by 2. Breaking it into two lots keeps things tidy.
Multiply by x
$x^3 + 3x^2 + x$
When you multiply powers of x you add the indices, so x times x squared is x cubed.
Multiply by 2
$2x^2 + 6x + 2$
Multiply each term inside the bracket by 2.
Add the two results
$x^3 + 3x^2 + x + 2x^2 + 6x + 2$
Write all the terms together, ready to collect like terms.
Collect like terms
$x^3 + 5x^2 + 7x + 2$
3x squared plus 2x squared is 5x squared, and x plus 6x is 7x.

Answer

x^3 + 5x^2 + 7x + 2

Question 4

2 markseasy

Expand and simplify

(x-4)^2

Worked solution

Write the square as a product
$(x-4)(x-4)$
Squaring a bracket means multiplying the bracket by itself. Never just square each term separately.
Multiply each pair of terms
$x^2 - 4x - 4x + 16$
Multiply out using FOIL. Note that negative four times negative four is positive sixteen.
Collect like terms
$x^2 - 8x + 16$
The two lots of -4x combine to -8x. This is the standard perfect-square pattern.

Answer

x^2 - 8x + 16

Question 5

2 markseasy

Factorise

x^2 + 7x + 12

Worked solution

Decide what we need
$(x+\square)(x+\square)$
Factorising is the reverse of expanding. We need two numbers that multiply to +12 and add to +7.
Find the number pair
$3 \times 4 = 12,\quad 3+4 = 7$
List factor pairs of 12 and check which pair adds to 7. Three and four work.
Write the factorised form
$(x+3)(x+4)$
Put the two numbers into the brackets. You can expand to check it returns the original.

Answer

(x+3)(x+4)

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