Question

The polynomial p{x) = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the remainder when p(x) is divided by x+ 2.

Answer 1

a = 5

Remainder when p(x) is divided by x+2 = 62

Explanation:

Given:

P(x) = x⁴-2x³+3x²-ax+3a-7

When x+1 divides the polynomial p(x) the ramainder is 19.

Applying remainder theorem,

x = -1

p(-1) = 19

Substitute the x = -1 into the polynomial expression

p(-1) = (-1)⁴-2(-1)³+3(-1)²-a(-1)+3a-7 = 19

1+2+3+a+3a-7 = 19

6-7+4a = 19

4a-1 = 19

4a = 19+1

4a = 20

a = 20/4

a = 5.

Hence, a = 5

p(x) = x⁴-2x³+3x²-5x+8

If p(x) is divided by x+2,

Then the remainder is p(-2)

p(-2) = (-2)⁴-2(-2)³+3(-2)²-5(-2)+8

p(-2) = 16+16+12+10+8

p(-2) = 62

Hence the remaider when p(x) is divided by x+2 is 62