Polynomials (Edexcel International A Level Maths: Pure 2)

Exam Questions

3 hours29 questions
1
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2 marks

Use the factor theorem to verify that left parenthesis x minus 2 right parenthesisĀ is a factor of x cubed minus x squared minus 14 x plus 24.

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2
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4 marks

DivideĀ  x cubed plus 6 x squared plus 11 x plus 6Ā  byĀ  open parentheses x plus 2 close parentheses.

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3
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6 marks

GivenĀ straight f left parenthesis x right parenthesis equals x cubed minus 5 x squared minus 2 x plus 24Ā has a root at x equals 3, fully factorise straight f left parenthesis x right parenthesis.

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4
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4 marks

Use polynomial division to show thatĀ left parenthesis x minus 2 right parenthesis is a factor of 2 x cubed plus 3 x squared minus 18 x plus 8.

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5
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4 marks

Given x equals negative 2 is a root of the functionĀ straight f left parenthesis x right parenthesis equals x cubed minus 3 x squared minus 8 x plus 4, fully factorise straight f left parenthesis x right parenthesis.

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6
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3 marks

Given that left parenthesis x minus 4 right parenthesis is a factor ofĀ x cubed minus k x squared minus 4 x plus 16, find the value of k.

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7
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4 marks

Given that x equals 1 half is a root of the functionĀ straight f left parenthesis x right parenthesis equals 2 x cubed plus left parenthesis p squared plus 1 right parenthesis x squared minus 11 x plus 4, find the possible values of p.

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8a
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4 marks
Given that left parenthesis x plus 1 right parenthesis is a factor ofĀ  straight f left parenthesis x right parenthesis equals x cubed minus 5 x squared plus 3 x plus 9, fully factoriseĀ straight f left parenthesis x right parenthesis.
8b
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3 marks
Sketch the graph of y equals straight f left parenthesis x right parenthesis, labelling the coordinates of all points where the graph intersects the coordinate axes.

Ā 

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9
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4 marks

Find the remainder whenĀ  x cubed minus 7 x minus 6Ā  is divided byĀ left parenthesis x plus 4 right parenthesis.

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10
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5 marks

The function straight f left parenthesis x right parenthesisĀ is given by straight f left parenthesis x right parenthesis equals x to the power of 4 plus a x cubed minus 13 x squared minus 38 x minus 4 b commaĀ where a and b are constants.

Given that both left parenthesis x plus 1 right parenthesis space space a n d space left parenthesis x plus 2 right parenthesisĀ are factors of straight f left parenthesis x right parenthesis comma find the valuesĀ a of bĀ .

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1
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2 marks

DivideĀ x cubed minus 6 x squared minus 9 x plus 14 Ā by left parenthesis x minus 7 right parenthesis.

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2a
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2 marks

Ā straight f left parenthesis x right parenthesis equals 2 x cubed minus x squared minus 16 x plus 15

Find the remainder when straight f left parenthesis x right parenthesis is divided by left parenthesis x minus 2 right parenthesis.

2b
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4 marks

Given that left parenthesis x plus 3 right parenthesisĀ is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesisĀ completely.

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3a
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2 marks

Ā straight f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared minus 72 x minus 35Ā 

Show that straight f left parenthesis x right parenthesis equals left parenthesis 2 x plus 1 right parenthesis left parenthesis a x squared plus b x plus c right parenthesis, Ā where a comma space band cĀ are constants to be found.
3b
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4 marks
Hence factorise straight f left parenthesis x right parenthesisĀ completely.
3c
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2 marks
Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.Ā 

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4a
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2 marks

straight f left parenthesis x right parenthesis equals 4 x cubed plus 4 x squared minus 23 x minus 30Ā 

Use the factor theorem to show that left parenthesis x plus 2 right parenthesisĀ is a factor of straight f left parenthesis x right parenthesis.
4b
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4 marks

Factorise straight f left parenthesis x right parenthesisĀ completely.

4c
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2 marks

Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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5a
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6 marks

straight f left parenthesis x right parenthesis equals x cubed plus 9 x squared plus r x plus s. Given that straight f left parenthesis 2 right parenthesis equals 0 andĀ straight f left parenthesis negative 1 right parenthesis equals negative 54 :

find the values ofĀ  r andĀ  s.

5b
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3 marks

Factorise straight f left parenthesis x right parenthesis completely.

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6
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5 marks

Consider the functionĀ f left parenthesis x right parenthesis equals 4 x cubed plus 6 x squared minus 7 x plus 2.

(i)
Find the quotient and remainder when 4 x cubed plus 6 x squared minus 7 x plus 2 is divided by left parenthesis x minus 2 right parenthesis.Ā 

(ii)
Hence writeĀ  4 x cubed plus 6 x squared minus 7 x plus 2Ā in the form left parenthesis x minus 2 right parenthesis left parenthesis a x squared plus b x plus c right parenthesis plus d comma whereĀ a comma space b comma space c and dĀ are constants to be determined.

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7a
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5 marks

The functionĀ f left parenthesis x right parenthesis equals 2 x cubed minus 5 x squared plus a x plus bĀ  has left parenthesis 2 x plus 3 right parenthesisĀ as a factor, and when f left parenthesis x right parenthesisĀ is divided by left parenthesis x minus 2 right parenthesis the remainder is 7.Ā 

Show that a andĀ b must satisfy the simultaneous equations: Ā 
Ā Ā Ā Ā Ā Ā Ā Ā Ā 2 a plus b equals 11

Ā Ā Ā Ā Ā Ā Ā Ā Ā 3 a minus 2 b equals negative 36

7b
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2 marks
Hence find a andĀ b.

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1
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2 marks

Divide x cubed minus 19 x minus 30Ā by left parenthesis x minus 5 right parenthesis.

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2a
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2 marks

straight f left parenthesis x right parenthesis equals x cubed minus 28 x plus 48

Find the remainder when straight f left parenthesis x right parenthesisĀ is divided by left parenthesis x minus 3 right parenthesis.
2b
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4 marks

Given that left parenthesis x plus 6 right parenthesisĀ is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesisĀ completely.

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3a
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2 marks

straight f left parenthesis x right parenthesis equals 6 x cubed minus 19 x squared plus 11 x plus 6

Show that straight f left parenthesis x right parenthesis equals left parenthesis 2 x minus 3 right parenthesis left parenthesis a x squared plus b x plus c right parenthesis where a comma space bĀ and c are constants to be found.Ā 
3b
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4 marks
Hence factorise straight f left parenthesis x right parenthesisĀ completely.
3c
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2 marks
Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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4a
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2 marks

straight f left parenthesis x right parenthesis equals 4 x cubed minus 7 x minus 3

Use the factor theorem to show that left parenthesis 2 x plus 1 right parenthesisĀ is a factor of straight f left parenthesis x right parenthesis.
4b
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4 marks
Factorise straight f left parenthesis x right parenthesis completely.
.
4c
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2 marks
Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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5a
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6 marks

straight f left parenthesis x right parenthesis equals x cubed plus r x squared plus s x minus 30. Given thatĀ straight f left parenthesis 2 right parenthesis equals 0 Ā andĀ straight f left parenthesis negative 3 right parenthesis equals negative 240 :

Ā find the values of rĀ and begin mathsize 20px style s end style.
5b
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3 marks
Factorise straight f left parenthesis x right parenthesis completely.

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6
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6 marks

Given that open parentheses x plus 4 close parentheses is a factor of the functionĀ f left parenthesis x right parenthesis equals p x cubed plus left parenthesis 5 p plus 1 right parenthesis x squared plus 5 q x minus 2 q minus 2 and that the remainder when f open parentheses x close parentheses Ā is divided by open parentheses x plus 1 close parenthesesĀ is negative 12, Ā find the values of the constants p and q.

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7
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4 marks

Show that 3 x cubed plus 16 x squared minus 22 x can be written in the formĀ left parenthesis 3 x plus 1 right parenthesis left parenthesis a x squared plus b x plus c right parenthesis plus d comma where a comma space b comma space cĀ and dĀ are constants to be found.

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1
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3 marks

Divide 4 x to the power of 4 minus 37 x squared plus 9Ā by left parenthesis 2 x minus 1 right parenthesis.

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2a
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2 marks

straight f left parenthesis x right parenthesis equals 6 x to the power of 4 plus 7 x cubed minus 27 x squared minus 28 x plus 12

Find the remainder when straight f left parenthesis x right parenthesisĀ is divided by left parenthesis 2 x plus 3 right parenthesis.
2b
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5 marks
Given that left parenthesis x plus 2 right parenthesis is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesis completely.Ā 

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3a
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2 marks

straight f left parenthesis x right parenthesis equals 3 x to the power of 4 plus x cubed minus 12 x squared minus 49 x minus 15

Show that straight f left parenthesis x right parenthesis equals left parenthesis 3 x plus 1 right parenthesis left parenthesis a x cubed plus b x squared plus c x plus d right parenthesis where a, b, cĀ and dĀ are constants to be found.

3b
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5 marks
Given that left parenthesis x minus 3 right parenthesisĀ is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesis completely.
3c
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2 marks
Hence show that the equation straight f left parenthesis x right parenthesis equals 0Ā has exactly 2 real roots.

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4
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4 marks

Ā Given that 3 is a root of the equation 2 x cubed minus x squared minus 11 x minus 12 equals 0, prove that the equation has no other real roots.

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5a
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1 mark

straight f left parenthesis x right parenthesis equals 2 x to the power of 4 minus 15 x cubed minus 10 x squared plus 105 x plus 98

Show that straight f left parenthesis negative 1 right parenthesis equals 0Ā and straight f left parenthesis negative 2 right parenthesis equals 0.
5b
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7 marks
Hence, solve straight f left parenthesis x right parenthesis equals 0.

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