C) The polynomial p(x) = 4x^4 + ax^3 + bx^2 – x + 2 is exactly divisible by 2x^2 – X – 1. Find the value of a and b.

a=4 , b=-9

Step-by-step explanation:

p(x) = 4x^4 + ax^3 + bx^2 -x +2 = (2x^2 - x - 1) q(x)

q(x) = cx^2 + dx + e

then we want to figure out the values of c, d and e

for c, 2x^2 * cx^2 = 4x^2, c=2

for e, -1 * e = 2, e = -2

for d, dx*(-1) + (-x) * e = -x, -dx + 2x = -x, d = 3

so,

q(x) = 2x^2 + 3x -2

to get the value of a and b, we just have to expand  

(2x^2 - x - 1) (2x^2 + 3x -2)

that is,

4x^4 + 6x^3 - 4x^2 - 2x^3 - 3x^2 + 2x - 2x^2 - 3x +2

simplify it,

p(x) = 4x^4 + 4x^3 - 9x^2 - x + 2

so a = 4 and b = -9

a = 4    b = -9

Step-by-step explanation:

2x² – X – 1 = (2x+1)(x-1)   2 roots: 1 and -1/2

if x=1:   4x⁴ + ax³ + bx² – x + 2 = 4(1)⁴+a(1)³+b(1)²-1+2 = 0

a + b + 5 = 0   ...(1)

if x = 1/2    4(1/2)⁴+a(1/2)³+b(1/2)²-1/2+2 = 0

14/8 + a/8 + b/4 = 0      ... times 8 both side

a + 2b + 14 = 0   ..(2)

(2) - (1): b + 9 = 0       b = -9

a = 4

check: 4x⁴ + 4x³ -9x² – x + 2 = (2 x - 1) (2 x + 1) (x - 1) (x + 2)

                                               = (2x² - x - 1) (2x² +x - 2)

i think the answer is d im not sure