What must be added to 12x 4 8x 3 23x 2 3x + 2 so that the resultant is divisible by 3x + 2 2x 5

CHAPTER 8

(i) Let f(x) = x3 - 2x2 - 9x + 18

x - 2 = 0

= (2)3 - 2(2)2 - 9(2) + 18

= 8 - 8 - 18 + 18

= 0

Hence, (x - 2) is a factor of f(x).

Now, we have:

(ii) Let f(x) = 2x3 + 5x2 - 28x - 15

x + 5 = 0

= 2(-5)3 + 5(-5)2 - 28(-5) - 15

= -250 + 125 + 140 - 15

= -265 + 265

= 0

Hence, (x + 5) is a factor of f(x).

Now, we have:

= (x + 5) [2x2 - 6x + x - 3]

= (x + 5) [2x(x - 3) + 1(x - 3)]

= (x + 5) (2x + 1) (x - 3)

(iii) Let f(x) = 3x3 + 2x2 - 3x - 2

3x + 2 = 0

Hence, (3x + 2) is a factor of f(x).

Now, we have:

(i)

(ii) Let f(x) = 2x3 + x2 - 13x + 6

For x = 2,

f(x) = f(2) = 2(2)3 + (2)2 - 13(2) + 6 = 16 + 4 - 26 + 6 = 0

Hence, (x - 2) is a factor of f(x).

(iii) f(x) = 3x3 + 2x2 - 23x - 30

For x = -2,

f(x) = f(-2) = 3(-2)3 + 2(-2)2 - 23(-2) - 30

= -24 + 8 + 46 - 30 = -54 + 54 = 0

Hence, (x + 2) is a factor of f(x).

(iv) f(x) = 4x3 + 7x2 - 36x - 63

For x = 3,

f(x) = f(3) = 4(3)3 + 7(3)2 - 36(3) - 63

= 108 + 63 - 108 - 63 = 0

Hence, (x + 3) is a factor of f(x).

(v) f(x) = x3 + x2 - 4x - 4

For x = -1,

f(x) = f(-1) = (-1)3 + (-1)2 - 4(-1) - 4

= -1 + 1 + 4 - 4 = 0

Hence, (x + 1) is a factor of f(x).

Let f(x) = 3x3 + 10x2 + x - 6

For x = -1,

f(x) = f(-1) = 3(-1)3 + 10(-1)2 + (-1) - 6 = -3 + 10 - 1 - 6 = 0

Hence, (x + 1) is a factor of f(x).

Now, 3x3 + 10x2 + x - 6 = 0

f (x) = 2x3 - 7x2 - 3x + 18

For x = 2,

f(x) = f(2) = 2(2)3 - 7(2)2 - 3(2) + 18

= 16 - 28 - 6 + 18 = 0

Hence, (x - 2) is a factor of f(x).

f(x) = x3 + 3x2 + ax + b

Since, (x - 2) is a factor of f(x), f(2) = 0

Since, (x + 1) is a factor of f(x), f(-1) = 0

Subtracting (ii) from (i), we get,

3a + 18 = 0

Substituting the value of a in (ii), we get,

b = a - 2 = -6 - 2 = -8

Now, for x = -1,

f(x) = f(-1) = (-1)3 + 3(-1)2 - 6(-1) - 8 = -1 + 3 + 6 - 8 = 0

Hence, (x + 1) is a factor of f(x).

Let f(x) = 4x3 - bx2 + x - c

It is given that when f(x) is divided by (x + 1), the remainder is 0.

4(-1)3 - b(-1)2 + (-1) - c = 0

-4 - b - 1 - c = 0

b + c + 5 = 0 ...(i)

It is given that when f(x) is divided by (2x - 3), the remainder is 30.

Multiplying (i) by 4 and subtracting it from (ii), we get,

5b + 40 = 0

b = -8

Substituting the value of b in (i), we get,

c = -5 + 8 = 3

Therefore, f(x) = 4x3 + 8x2 + x - 3

Now, for x = -1, we get,

f(x) = f(-1) = 4(-1)3 + 8(-1)2 + (-1) - 3 = -4 + 8 - 1 - 3 = 0

Hence, (x + 1) is a factor of f(x).

f(x) = x2 + px + q

It is given that (x + a) is a factor of f(x).

g(x) = x2 + mx + n

It is given that (x + a) is a factor of g(x).

From (i) and (ii), we get,

pa - q = ma - n

n - q = a(m - p)

Hence, proved.

Let f(x) = ax3 + 3x2 - 3

When f(x) is divided by (x - 4), remainder = f(4)

f(4) = a(4)3 + 3(4)2 - 3 = 64a + 45

Let g(x) = 2x3 - 5x + a

When g(x) is divided by (x - 4), remainder = g(4)

g(4) = 2(4)3 - 5(4) + a = a + 108

It is given that f(4) = g(4)

64a + 45 = a + 108

63a = 63

a = 1

Let f(x) = x3 - ax2 + x + 2

It is given that (x - a) is a factor of f(x).

a3 - a3 + a + 2 = 0

a + 2 = 0

a = -2

Let the number to be subtracted from the given polynomial be k.

Let f(y) = 3y3 + y2 - 22y + 15 - k

It is given that f(y) is divisible by (y + 3).

3(-3)3 + (-3)2 - 22(-3) + 15 - k = 0

-81 + 9 + 66 + 15 - k = 0

9 - k = 0

k = 9