Name:
Level 2 Further Maths
Factor Theorem
Ensure you have: Pencil or pen, a calculator
Guidance
1. Read each question carefully before you begin answering it.
2. Check your answers seem right.
3. Always show your workings
Revision for this topic
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1. Use factor theorem to show that (x − 1) is a factor of x 3 − 3x 2 − 13x + 15
(1)
2. Use factor theorem to show that (x − 3) is a factor of x 3 − 10x 2 + 21x
(1)
3. Use factor theorem to show that (x + 4) is a factor of x 3 + 4x 2 − 3x − 12
(1)
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4. Use factor theorem to show that (2x − 1) is a factor of 2x 3 + 7x 2 + 2x − 3
(2)
5. f (x) = 4x 3 + 5x 2 − 23x − 6
Use the factor theorem to show that (4x + 1) is a factor of f (x)
(2)
6. Use the factor theorem to show that (x + 5)
is not a factor of x 3 − 12x 2 + 47x − 35
(2)
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7. (a) Use the factor theorem to show that (x − 1)
is a factor of x 3 − x 2 − 4x + 4
(1)
(b) Hence, factorise fully x 3 − x 2 − 4x + 4
……………………………………
(3)
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8. (a) Use the factor theorem to show that (x − 2)
is a factor of x 3 − 9x 2 + 20x − 12
(1)
(b) Hence, factorise fully x 3 − 9x 2 + 20x − 12
……………………………………
(3)
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9. (a) Use the factor theorem to show that (x + 4)
is a factor of 2x 3 + 5x 2 − 14x − 8
(1)
(b) Hence, factorise fully 2x 3 + 5x 2 − 14x − 8
……………………………………
(4)
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10. (a) Use the factor theorem to show that (2x − 3)
is a factor of 2x 3 + x 2 − 12x + 9
(2)
(b) Hence, factorise fully 2x 3 + x 2 − 12x + 9
……………………………………
(3)
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11. (a) Use the factor theorem to show that (x − 2) and (x + 5)
are factors of x 3 + 2x 2 − 13x + 10
(2)
(b) Use the factor theorem to show that (x − 2) and (x + 5)
are also factors of x 3 + 11x 2 + 14x − 80
(2)
3 2
x + 2x − 13x + 10
(c) Hence, simplify fully
x 3 + 11x 2 + 14x − 80
……………………………………
(3)
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12. (a) Show that (x + 3) is a factor of x 3 + 3x 2 − 49x − 147
(2)
(b) Hence, or otherwise, find all the solutions of x 3 + 3x 2 − 49x − 147 = 0
……………………………………
(4)
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13. Factorise fully x 3 − 6x 2 + 11x − 6
(5)
14. (x − 5) is a factor of x 3 − x 2 − 32x + a
Work out the value of a
a = …………………………
(2)
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14. (x + 4) is a factor of x 3 + 11x 2 + a x − 72
Work out the value of a
a = …………………………
(3)
15. Given (x − 1) is a factor of 3x 3 − 15x 2 + a x + a
Find the value of a
a = …………………………
(4)
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16. (x + a) is a factor of x 3 − 7x 2 + a x + 20a
(a) Show that a=2
(2)
(b) Solve x 3 − 7x 2 + 2x + 40 = 0
……………………………………
(4)
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17. Below is the graph of y = 5x 3 + 23x 2 − 40x + 12
Find the coordinates of the points a and b, where the graph of
y = 5x 3 + 23x 2 − 40x + 12 crosses the x-axis.
……………………………………
(4)
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18. Solve x 3 − 19x 2 + 103x − 165 = 0
……………………………………
(5)
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