Polynomials (OCR AS Maths: Pure)

Exam Questions

3 hours36 questions
1
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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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2
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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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3
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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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4
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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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5
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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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6
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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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7
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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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8
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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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9
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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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10a
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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.

10b
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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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11
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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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12
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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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1 mark

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

1b
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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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2
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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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3
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2 marks

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

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4
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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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5a
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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.

5b
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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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6a
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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.

6b
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4 marks

Hence factorise straight f left parenthesis x right parenthesis spacecompletely.

6c
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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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7a
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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.

7b
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4 marks

Factorise space straight f left parenthesis x right parenthesis completely.

7c
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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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8a
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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.

8b
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3 marks

Factorise space straight f left parenthesis x right parenthesis completely.

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1a
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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.

1b
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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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2
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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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3
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2 marks

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

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4
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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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5a
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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.

5b
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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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6a
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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.

6b
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4 marks

Hence factorise straight f left parenthesis x right parenthesis completely.

6c
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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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7a
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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.

7b
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4 marks

Factorise straight f left parenthesis x right parenthesis completely.

7c
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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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8a
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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.

8b
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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

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.

1b
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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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2
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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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3
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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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4
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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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5a
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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.

5b
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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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6a
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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.

6b
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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.

6c
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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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7
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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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8a
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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.

8b
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7 marks

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

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