A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 7: Algebraic methods · Lesson 1

Algebraic fractions and dividing polynomials

73 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-algebraic-fractions-and-dividing-polys/

Date

Do Now

Quick questions to start.

A1
Needed today
Factorise $5 x^{2} - 13 x - 6$ .
[2 marks]
Answer
A2
Needed today
Work out $9437 \div 26$ , giving the remainder.
[2 marks]
Answer $\dots\dots$ remainder $\dots\dots$
A3
Last lesson
Find the tangent to $x^{2} + y^{2} = 13$ at $(2, - 3)$ .
[2 marks]
Answer
A4
From chapter 1
Simplify $(12 x^{5} - 8 x^{3}) \div 4 x^{2}$ .
[2 marks]
Answer

A-Level

Work through each example, then try the one beside it.

Example 1

Simplify fully $\frac{3 x ^{2} + 7 x - 6}{x ^{2} - 9}$ .

Hint Factorise the top and the bottom. You may only cancel a bracket that is a factor of the WHOLE top and the WHOLE bottom.

B1
Simplify fully $\frac{2 x ^{2} - 3 x - 20}{4 x ^{2} - 25}$ .
[3 marks]
Answer

Example 2

Divide $2 x^{3} - 3 x^{2} - 11 x + 6$ by $(x + 2)$ , and hence factorise $2 x^{3} - 3 x^{2} - 11 x + 6$ completely.

Hint Divide the first term by

x

, multiply

(x + 2)

by what you get, subtract, bring down the next term, and repeat.

B2
Divide $3 x^{3} + 8 x^{2} - 15 x + 4$ by $(x + 4)$ . Hence solve $3 x^{3} + 8 x^{2} - 15 x + 4 = 0$ .
[5 marks]
Answer $x = \dots\dots$ , $x = \dots\dots$ , $x = \dots\dots$

Example 3

Find the quotient and the remainder when $4 x^{3} - 10 x^{2} + 5$ is divided by $(2 x + 1)$ .

Hint There is no

x

term: write

4 x^{3} - 10 x^{2} + 0 x + 5

so that every power has its own column. Each step divides by

2 x

B3
Find the quotient and the remainder when $6 x^{4} - 5 x^{3} + 10 x^{2} + 3$ is divided by $(3 x - 1)$ .
[5 marks]
Answerquotient $\dots\dots$ , remainder $\dots\dots$

A-Level

C1
Use algebraic division to show that $(x + 3)$ is a factor of $3 x^{3} + x^{2} - 20 x + 12$ . Hence solve $3 x^{3} + x^{2} - 20 x + 12 = 0$ .
[5 marks]
Answer $x = \dots\dots$ , $x = \dots\dots$ , $x = \dots\dots$

C2
Given that $(x + 4)$ is a factor of $x^{3} + 3 x^{2} - 6 x - 8$ , simplify fully $\frac{x ^{3} + 3 x ^{2} - 6 x - 8}{x ^{2} + 6 x + 8}$ .
[6 marks]
Answer
C3
Divide $8 x^{3} + 27$ by $(2 x + 3)$ .
[3 marks]
Answer

Edexcel exam style

D1
Simplify fully $\frac{x ^{2} + 5 x - 24}{x ^{2} - 9}$ . Hence solve the equation $\frac{x ^{2} + 5 x - 24}{x ^{2} - 9} = 3$ .
[6 marks]
Answer $\dots\dots$ ; $x = \dots\dots$

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify an algebraic fraction by factorising and cancelling common factors.
I can divide a polynomial by a linear expression, writing any missing terms as zero.
I can find the remainder, and use division to factorise a cubic completely.
I can use the quotient to solve a cubic or show how many real roots it has.

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