Question

If polynomial p(y)=-2y4+8y3-3y2+a2-9y+4 leaves remainder 32 when divided by (u-2). find the value of a. hence find remainder when p(y) is divided by (y+2).

Answer 1

$P(2) = - 2 *2^4+8*2^3-3*2^2+a^2-9*2+4 = 32 \\ \\ -32+64-12+a^2-18+4 =32 \\ \\ a^2 = 26 \\ \\ a = +- \sqrt(26) \\ \\ P(-2) = -2 * (-2)^4 + 8 * (-2)^3 - 3*(-2)^2 + 26 - 9*(-2) + 4 \\ \\ \ \ \ \ \ \ = -32 -64 -12 + 26+18 + 4 \\ \\ \ \ \ \ \ = \ -60$

Answer 2

Assuming that, the polynomial P(y) is divided by (y-2) not by (u-2) as you mentioned.

$P(2) = -2(2)^4 + 8(2)^3 -3(2)^2 +a^2 -9(2) + 4$

$P(2) = -2(16) + 8(8) -3(4) +a^2 -(18) + 4$

$P(2) = -32 + 64 - 12 +a^2 - 14$

$P(2) = 6 +a^2$

But the remainder = 32

$a^2 + 6 = 32$
$a^2 = 26$
$a = \sqrt{26}$

Now, the remainder when P(x) divided by (y+2)

$P(-2) = -2(-2)^4 + 8(-2)^3 -3(-2)^2 +( \sqrt{26})^2 -9(-2) + 4$
 
$P(-2) = -2(16) + 8(-8) -3(4) + 26 +18 + 4$
 
$P(-2) = -32 - 64 - 12 + 26 + 32$
$P(-2) = -76 + 26$ 
$P(-2) = -50$