Factors of Polynomials (CIE IGCSE Additional Maths)

Exam Questions

58 mins8 questions
1a
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2 marks

straight p left parenthesis x right parenthesis space equals space 15 x cubed space plus 22 x squared space minus 15 x plus 2

Find the remainder when straight p left parenthesis x right parenthesis is divided by x plus 1.

1b
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4 marks
(i)
Show that x plus 2 is a factor of straight p left parenthesis x right parenthesis .

[1]

(ii)
Write straight p left parenthesis x right parenthesis as a product of linear factors.

[3]

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1a
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4 marks

The polynomial space straight p left parenthesis x right parenthesis space equals space 6 x cubed space plus space a x squared space plus space b x space plus space 2 space, where a and b are integers, has a factor of x space – space 2 .
Given that space straight p left parenthesis 1 right parenthesis space equals space – 2 straight p left parenthesis 0 right parenthesis, find the values of a and b.

1b
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4 marks

Using your values of a and b,

(i)
find the remainder when straight p left parenthesis x right parenthesis is divided byspace 2 x space – space 1

(ii)
factorise straight p left parenthesis x right parenthesis.

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2a
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5 marks

straight p left parenthesis x right parenthesis space equals space a x cubed space plus 3 x squared space plus b x minus 12 has a factor of 2 x plus 1. When straight p left parenthesis x right parenthesis is divided by x minus 3 the remainder is 105.

Find the value of a and of b.

2b
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2 marks

Using your values of a and b, write straight p left parenthesis x right parenthesis space as a product of 2 x plus 1 and a quadratic factor.

2c
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2 marks

Hence solvespace straight p left parenthesis x right parenthesis space= 0.

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3a
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6 marks

straight p left parenthesis x right parenthesis space equals space 6 x cubed space plus a x squared space plus 12 x plus b, where a and b are integers.



straight p left parenthesis x right parenthesis has a remainder of 11 when divided by x minus 3 and a remainder of negative 21 when divided by x plus 1.
Given that straight p left parenthesis x right parenthesis space equals space left parenthesis x minus 2 right parenthesis Q left parenthesis x right parenthesis, find Q left parenthesis x right parenthesis, a quadratic factor with numerical coefficients.

3b
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2 marks

Hence solve straight p left parenthesis x right parenthesis space equals space 0.

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4
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5 marks

The three roots of straight p open parentheses x close parentheses equals 0 , where straight p open parentheses x close parentheses equals 2 x cubed plus a x squared plus b x plus c are x equals 1 half comma space x equals n and x equals negative n, where a comma space b comma space c and n are integers. The y-intercept of the graph of y space equals space straight p left parenthesis x right parenthesis is 4. Find straight p left parenthesis x right parenthesis, simplifying your coefficients.

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1a
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4 marks

The polynomial straight p left parenthesis x right parenthesis space equals space a x cubed space minus 9 x squared space plus b x minus 6, where a and b are constants, has a factor of x minus 2. The polynomial has a remainder of 66 when divided by x minus 3.

Find the value of a and of b.

1b
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2 marks

Using your values of a and b, show that straight p left parenthesis x right parenthesis space equals space left parenthesis x minus 2 right parenthesis straight q left parenthesis x right parenthesis, where straight q left parenthesis x right parenthesis is a quadratic factor to be found.

1c
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2 marks

Hence show that the equation straight p left parenthesis x right parenthesis space equals space 0 has only one real solution.

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2a
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1 mark

straight p left parenthesis x right parenthesis space equals space 2 x cubed space minus 3 x squared space minus 23 x plus 12

Find the value of straight p open parentheses 1 half close parentheses.

2b
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5 marks

Write straight p left parenthesis x right parenthesis subscript blank as the product of three linear factors and hence solve straight p left parenthesis x right parenthesis space equals space 0.

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3a
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6 marks

The polynomial straight p left parenthesis x right parenthesis space equals space a x cubed space plus b x squared space minus 19 x plus 4, where a and b are constants, has a factor x plus 4 and is such that 2 straight p left parenthesis 1 right parenthesis space equals space 5 straight p left parenthesis 0 right parenthesis.

Show that straight p left parenthesis x right parenthesis space equals space left parenthesis x plus 4 right parenthesis left parenthesis A x squared space plus B x plus C right parenthesis, where AB and C are integers to be found.

3b
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1 mark

Hence factorise straight p left parenthesis x right parenthesis.

3c
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1 mark

Find the remainder when straight p apostrophe left parenthesis x right parenthesis is divided byspace x.

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