2024年5月1日发(作者：小米手环怎么调时间)

C2 A

LGEBRA

Worksheet A

1

Find the quotient obtained in dividing

a (x

3

+ 2x

2

− x − 2) by (x + 1)

c (20 + x + 3x

2

+ x

3

) by (x + 4)

e (6x

3

− 19x

2

− 73x + 90) by (x − 5)

g (x

3

− 2x + 21) by (x + 3)

b (x

3

+ 2x

2

− 9x + 2) by (x − 2)

d (2x

3

− x

2

− 4x + 3) by (x − 1)

f (−x

3

+ 5x

2

+ 10x − 8) by (x + 2)

h (3x

3

+ 16x

2

+ 72) by (x + 6)

2

Find the quotient and remainder obtained in dividing

a (x

3

+ 8x

2

+ 17x + 16) by (x + 5)

c (3x

3

+ 4x

2

+ 7) by (2 + x)

e (4x

3

+ 2x

2

− 16x + 3) by (x − 3)

b (x

3

− 15x

2

+ 61x − 48) by (x − 7)

d (−x

3

− 5x

2

+ 15x − 50) by (x + 8)

f (1 − 22x

2

− 6x

3

) by (x + 2)

3

Use the factor theorem to determine whether or not

a (x − 1) is a factor of (x

3

+ 2x

2

− 2x − 1)

c (x − 3) is a factor of (x

3

− x

2

− 14x + 27)

b (x + 2) is a factor of (x

3

− 5x

2

− 9x + 2)

d (x + 6) is a factor of (2x

3

+ 13x

2

+ 2x − 24)

4

e (2x + 1) is a factor of (2x

3

− 5x

2

+ 7x − 14) f (3x − 2) is a factor of (2 − 17x + 25x

2

− 6x

3

)

f(x) ≡ x

3

− 2x

2

− 11x + 12.

a Show that (x − 1) is a factor of f(x).

b Hence, express f(x) as the product of three linear factors.

5

g(x) ≡ 2x

3

+ x

2

− 13x + 6.

Show that (x + 3) is a factor of g(x) and solve the equation g(x) = 0.

6

f(x) ≡ 6x

3

− 7x

2

− 71x + 12.

Given that f(4) = 0, find all solutions to the equation f(x) = 0.

7

g(x) ≡ x

3

+ 7x

2

+ 7x − 6.

Given that x = −2 is a solution to the equation g(x) = 0,

a express g(x) as the product of a linear factor and a quadratic factor,

b find, to 2 decimal places, the other two solutions to the equation g(x) = 0.

8

f(x) ≡ x

3

+ 2x

2

− 11x − 12.

a Evaluate f(1), f(2), f(−1) and f(−2).

b Hence, state a linear factor of f(x) and fully factorise f(x).

9

By first finding a linear factor, fully factorise

a x

3

− 2x

2

− 5x + 6

d 3x

3

− 4x

2

− 35x + 12

b x

3

+ x

2

− 5x − 2

e x

3

+ 8

c 20 + 11x − 8x

2

+ x

3

f 12 + 29x + 8x

2

− 4x

3

10

Solve each equation, giving your answers in exact form.

a x

3

− x

2

− 10x − 8 = 0

d x

3

− 5x

2

+ 3x + 1 = 0

b x

3

+ 2x

2

− 9x − 18 = 0

e x

2

(x + 4) = 3(3x + 2)

c 4x

3

− 12x

2

+ 9x = 2

f x

3

− 14x + 15 = 0

11

C2 A

LGEBRA

f(x) ≡ 2x

3

− x

2

− 15x + c.

Worksheet A continued

Given that (x − 2) is a factor of f(x),

a find the value of the constant c,

b fully factorise f(x).

12

g(x) ≡ x

3

+ px

2

− 13x + q.

Given that (x + 1) and (x − 3) are factors of g(x),

a show that p = 3 and find the value of q,

b solve the equation g(x) = 0.

13

Use the remainder theorem to find the remainder obtained in dividing

a (x

3

+ 4x

2

− x + 6) by (x − 2)

c (2x

3

+ x

2

− 9x + 17) by (x + 5)

e (2x

3

− 3x

2

− 20x − 7) by (2x + 1)

b (x

3

− 2x

2

+ 7x + 1) by (x + 1)

d (8x

3

+ 4x

2

− 6x − 3) by (2x − 1)

f (3x

3

− 6x

2

+ 2x − 7) by (3x − 2)

14

15

16

Given that when (x

3

− 4x

2

+ 5x + c) is divided by (x − 2) the remainder is 5, find the value of the

constant c.

Given that when (2x

3

− 9x

2

+ kx + 5) is divided by (2x − 1) the remainder is −2, find the value of

the constant k.

Given that when (2x

3

+ ax

2

+ 13) is divided by (x + 3) the remainder is 22,

a find the value of the constant a,

b find the remainder when (2x

3

+ ax

2

+ 13) is divided by (x − 4).

f(x) ≡ px

3

+ qx

2

+ qx + 3.

17

Given that (x + 1) is a factor of f(x),

a find the value of the constant p.

Given also that when f(x) is divided by (x − 2) the remainder is 15,

b find the value of the constant q.

18

p(x) ≡ x

3

+ ax

2

+ 9x + b.

Given that (x − 3) is a factor of p(x),

a find a linear relationship between the constants a and b.

Given also that when p(x) is divided by (x + 2) the remainder is −30,

b find the values of the constants a and b.

19

f(x) ≡ 4x

3

− 6x

2

+ mx + n.

20

Given that when f(x) is divided by (x + 1) the remainder is 3 and that when f(x) is divided

by (2x − 1) the remainder is 15, find the values of the constants m and n.

g(x) ≡ x

3

+ cx + 3.

Given that when g(x) is divided by (x − 4) the remainder is 39,

a find the value of the constant c,

b find the quotient and remainder when g(x) is divided by (x + 2).