Polynomials (Edexcel A Level Maths: Pure): Exam Questions

4 hours44 questions
13 marks

straight f open parentheses x close parentheses equals a x cubed plus 10 x squared minus 3 a x minus 4

Given that left parenthesis x minus 1 right parenthesis is a factor of straight f left parenthesis x right parenthesis, find the value of the constant a.

You must make your method clear.

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23 marks

straight f open parentheses x close parentheses equals 3 x cubed plus 2 a x squared minus 4 x plus 5 a

Given that open parentheses x plus 3 close parentheses is a factor of straight f open parentheses x close parentheses, find the value of the constant a.

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3
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5 marks

Expand and simplify

(i) left parenthesis 2 x plus 3 right parenthesis left parenthesis x minus 4 right parenthesis 

(ii) 2 p left parenthesis p plus 3 right parenthesis left parenthesis p minus 2 right parenthesis 

(iii) left parenthesis y minus 1 right parenthesis left parenthesis y minus 2 right parenthesis squared 

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4
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2 marks

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

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5
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4 marks

Factorise

(i) 4 x squared minus 4 x minus 15 

(ii) 3 x cubed plus 11 x squared minus 4 x  

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6
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4 marks

Divide x cubed plus 6 x squared plus 11 x plus 6 by  left parenthesis x plus 2 right parenthesis.

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7
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6 marks

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

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8
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4 marks

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

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9
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4 marks

Given x equals negative 2 space is a root of the function 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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10
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3 marks

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

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11
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4 marks

Given that x equals 1 half is a root of the function 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 space p.

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12a
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4 marks

Given that left parenthesis x plus 1 right parenthesis space is a factor of straight f left parenthesis x right parenthesis equals x cubed minus 5 x squared plus 3 x plus 9, fully factorise straight f left parenthesis x right parenthesis.

12b
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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.

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13
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4 marks

Find the remainder when x cubed minus 7 x minus 6 is divided by left parenthesis x plus 4 right parenthesis.

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14
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5 marks

The function straight f left parenthesis x right parenthesis 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, where a and b are constants.

Given that both left parenthesis x plus 1 right parenthesis and left parenthesis x plus 2 right parenthesis are factors of straight f left parenthesis x right parenthesis find the values of a and b.

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1a
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2 marks

straight f open parentheses x close parentheses equals 2 x cubed plus 5 x squared plus 2 x plus 15

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

1b2 marks

Find the constants a, b and c such that

straight f open parentheses x close parentheses equals open parentheses x plus 3 close parentheses open parentheses a x squared plus b x plus c close parentheses

1c
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1 mark

Hence show that straight f open parentheses x close parentheses equals 0 has only one real root.

1d1 mark

Write down the real root of the equation straight f open parentheses x minus 5 close parentheses equals 0

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2
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3 marks

straight f left parenthesis x right parenthesis equals left parenthesis x minus 4 right parenthesis left parenthesis x squared minus 3 x plus k right parenthesis minus 42 where k is a constant

Given that left parenthesis x plus 2 right parenthesis is a factor of straight f left parenthesis x right parenthesis, find the value of k.

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3a
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1 mark

Expand and simplify space y left parenthesis 2 x plus 2 right parenthesis left parenthesis 7 minus x right parenthesis.

3b
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2 marks

A rectangle has side lengths of left parenthesis 3 x minus 2 y plus 5 right parenthesis units and left parenthesis x plus 3 y minus 1 right parenthesis units. Find an expression for the area of the rectangle in terms of x and space y.

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4
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2 marks

Given that left parenthesis 3 x plus y right parenthesis left parenthesis 2 x minus 3 y right parenthesis left parenthesis x minus 2 y right parenthesis equals a x cubed plus b x squared y plus c x y squared plus d y cubed, where a comma space b comma space c space and space d are constants, find the values of a comma space b comma space c space and space d.

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5
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2 marks

Factorise completely 3 x cubed minus 51 x squared plus 126 x.

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6
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2 marks

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

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7a
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2 marks

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 spaceis divided by left parenthesis x minus 2 right parenthesis.

7b
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4 marks

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

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8a
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2 marks

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

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 where a comma space b space and space c are constants to be found.

8b
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4 marks

Hence factorise straight f left parenthesis x right parenthesis spacecompletely.

8c
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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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9a
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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 30

Use the factor theorem to show that left parenthesis x plus 2 right parenthesis spaceis a factor of space straight f left parenthesis x right parenthesis.

9b
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4 marks

Factorise space straight f left parenthesis x right parenthesis completely.

9c
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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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10a
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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 spaceand straight f left parenthesis negative 1 right parenthesis equals negative 54:

find the values of r and s.

10b
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3 marks

Factorise space straight f left parenthesis x right parenthesis completely.

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1a2 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

straight f open parentheses x close parentheses equals 4 x cubed plus 5 x squared minus 10 x plus 4 a space space space space space space space space x element of straight real numbers

where a is a positive constant.

Given open parentheses x minus a close parentheses is a factor of straight f open parentheses x close parentheses, show that

a open parentheses 4 a squared plus 5 a minus 6 close parentheses equals 0

1b
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4 marks

Hence

(i) Find the value of a

(ii) use algebra to find the exact solutions of the equation

straight f open parentheses x close parentheses equals 3

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2a
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3 marks

straight f open parentheses x close parentheses equals negative 3 x cubed plus 8 x squared minus 9 x plus 10 comma space space space space space x element of straight real numbers

(i) Calculate straight f open parentheses 2 close parentheses

(ii) Write straight f open parentheses x close parentheses as a product of two algebraic factors.

2b
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2 marks

Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation

negative 3 y to the power of 6 plus 8 y to the power of 4 minus 9 y squared plus 10 equals 0

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3a
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1 mark

Expand and simplify left parenthesis 2 minus x right parenthesis left parenthesis 3 x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis.

3b
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2 marks

A square has side lengths of left parenthesis 5 x minus 2 y plus 3 right parenthesis units. Find an expression for the length of the diagonal of the square in terms of x and space y.

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4
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2 marks

Given that left parenthesis 2 x minus 3 y right parenthesis squared left parenthesis y minus 2 x right parenthesis equals a x cubed plus b x squared y plus c x y squared plus d y cubed, where a comma space b comma space c space and space d are constants, find the values of a comma space b comma space c and d.

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5
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2 marks

Factorise completely 15 x cubed plus 19 x squared minus 10 x.

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6
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2 marks

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

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7a
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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 parenthesis spaceis divided by left parenthesis x minus 3 right parenthesis.

7b
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4 marks

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

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8a
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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 b and c are constants to be found.

8b
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4 marks

Hence factorise straight f left parenthesis x right parenthesis completely.

8c
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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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9a
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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 parenthesis is a factor of straight f left parenthesis x right parenthesis.

9b
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4 marks

Factorise straight f left parenthesis x right parenthesis completely.

9c
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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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10a
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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 that straight f left parenthesis 2 right parenthesis equals 0 space and straight f left parenthesis negative 3 right parenthesis equals negative 240

find the values of r and s.

10b
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3 marks

Factorise straight f left parenthesis x right parenthesis completely.

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1a
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2 marks
Graph with two intersecting curves, C1 and C2, on an xy-plane. C1 is ascending in quadrant 1 only from just under half way up the y axis, while C2 is a downward (negative) parabola. C1 and C2 intersect twice
Figure 4

Figure 4 shows a sketch of part of the curve C subscript 1 with equation

y equals 2 x cubed plus 10 space space space space space space space space space space space space space space space space space space x greater than 0

and part of the curve C subscript 2 with equation

y equals 42 x minus 15 x squared minus 7 space space space space space space space space space space space space space space space space space space x greater than 0

Verify that the curves intersect at space x equals 1 half

1b5 marks

The curves intersect again at the point P

Using algebra and showing all stages of working, find the exact x coordinate of P

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2a2 marks

In this question you must show detailed reasoning.

Solutions relying on calculator technology are not acceptable.

The curve C subscript 1 has equation y equals 8 minus 10 x plus 6 x squared minus x cubed

The curve C subscript 2 has equation y equals x squared minus 12 x plus 14

Verify that when x equals 1 the curves C subscript 1 and C subscript 2 intersect.

2b
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5 marks

The curves also intersect when x equals k.

Given that k less than 0

use algebra to find the exact value of k.

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3a
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2 marks

Expand and simplify left parenthesis x plus y right parenthesis left parenthesis x minus y right parenthesis left parenthesis y minus x right parenthesis left parenthesis negative x minus y right parenthesis.

3b
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2 marks

A cuboid has a length of left parenthesis 2 x minus 3 y plus 3 right parenthesis units, a width of left parenthesis 2 x plus 3 y minus 3 right parenthesis units, and a height of left parenthesis x minus y right parenthesis units.  Find an expression for the volume of the cuboid in terms of x and space y.

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4
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3 marks

Given that left parenthesis a x plus b y right parenthesis left parenthesis 2 x plus y right parenthesis left parenthesis x minus 3 y right parenthesis equals 8 x cubed plus c x squared y plus d x y squared minus 9 y cubed, where a comma space b comma space c and d are constants, find the values of a comma space b comma space c and d.

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5
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3 marks

Factorise completely x to the power of 5 y minus x y to the power of 5.

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6
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3 marks

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

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7a
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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 parenthesis is divided by left parenthesis 2 x plus 3 right parenthesis.

7b
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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.

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8a
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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 comma space b comma space c and d are constants to be found.

8b
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5 marks

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

8c
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2 marks

Hence show that the equation straight f left parenthesis x right parenthesis equals 0 space has exactly 2 real roots.

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9
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4 marks

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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10a
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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 0 space and straight f left parenthesis negative 2 right parenthesis equals 0.

10b
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7 marks

Hence, solve straight f left parenthesis x right parenthesis equals 0.

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