GCSEFurther MathsAQAExam Questions2. Algebra & FunctionsPolynomials & Factor Theorem

Polynomials & Factor Theorem (AQA GCSE Further Maths): Exam Questions

Exam code: 8365

51 mins13 questions
Medium
Hard
Very Hard
1
Sme Calculator1 mark

Use the factor theorem to show that open parentheses x minus 3 close parentheses is a factor of x cubed minus 4 x squared plus x plus 6.

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2
Sme Calculator1 mark

Use the factor theorem to show that open parentheses x plus 2 close parentheses is a factor of x cubed minus 7 x minus 6

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3a
Sme Calculator1 mark

Use the factor theorem to show that left parenthesis x space minus space 2 right parenthesis is a factor of x cubed plus space 8 x squared plus 5 x minus 50

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3b
Sme Calculator3 marks

Hence, factorise fully x cubed plus 8 x squared plus 5 x minus 50.

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4
Sme Calculator3 marks

open parentheses x minus 1 close parentheses is a factor of x cubed plus 6 x squared plus 3 x minus 10.

Use this fact to factorise x cubed plus 6 x squared plus 3 x minus 10 fully.

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5
Sme Calculator3 marks

straight f open parentheses x close parentheses equals x cubed minus 10 x minus c where c is a positive integer.

left parenthesis x plus c right parenthesis is a factor of straight f left parenthesis x right parenthesis.

Use the factor theorem to work out the value of c.

How did you do?

1a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 3 x cubed – 2 x squared – 7 x – 2

Use the factor theorem to show that left parenthesis 3 x plus 1 right parenthesis is a factor of straight f open parentheses x close parentheses.

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1b
Sme Calculator3 marks

Factorise straight f open parentheses x close parentheses fully.

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2a
Sme Calculator2 marks

straight f open parentheses x close parentheses equals 200 x cubed plus 100 x squared minus 18 x minus 9

Use the factor theorem to show that open parentheses 2 x plus 1 close parentheses is a factor of straight f open parentheses x close parentheses.

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2b
Sme Calculator3 marks

Hence solve  straight f open parentheses x close parentheses equals 0.

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3a
Sme Calculator2 marks

open parentheses x minus 3 close parentheses is a factor of x cubed minus 8 x squared plus a x plus 42  where a is an integer.

Show that the value of a is 1.

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3b
Sme Calculator3 marks

Hence, factorise fully   x cubed minus 8 x squared plus x plus 42.

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4
Sme Calculator5 marks

Solve x cubed plus x squared minus 26 x plus 24 equals 0

You must show your working.

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5
Sme Calculator4 marks

The curve y equals 2 x cubed minus x squared minus 32 x plus 16 has an x-intercept of 1 half.

Use this information to find the other two x-intercepts.

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1
Sme Calculator5 marks

straight f open parentheses x close parentheses equals x cubed plus p x squared plus q where p and q are integers.

The equation straight f open parentheses x close parentheses equals 0 has the solution x equals 2 and a different non-zero solution that is a repeated solution.

By factorising straight f open parentheses x close parentheses, or otherwise, find the values of p and q.

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2
Sme Calculator5 marks

By first finding a factor of x cubed minus x squared plus 3 x minus 10, show that the equation x cubed minus x squared plus 3 x minus 10 equals 0 has one solution only.

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3a
Sme Calculator2 marks

straight f open parentheses x close parentheses equals a x cubed minus b x squared minus a cubed x plus a squared b where a and b are non-zero integers.

Work out straight f open parentheses b over a close parentheses.

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3b
Sme Calculator3 marks

Factorise straight f open parentheses x close parentheses fully, giving your answer in terms of a and b.

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