Question

If a polynomial 8x4-8x3-18x2-px-q is exactly divisible by 4x2-4x+1 find value of p and q
Can anyone do this without long division....
Then I will mark you as BRAINLIEST

Answer 1

Given,

8x⁴ - 8x³ - 18x² - px - q is exactly divisible by 4x² - 4x + 1.

To find,

The value of p and q.

8x⁴ - 8x³ - 18x² - px - q is exactly divisible by 4x² - 4x + 1.

⇒4x² - 4x + 1 is a factor of 8x⁴ - 8x³ - 18x² - px - q.

⇒(2x - 1)² is a factor of 8x⁴ - 8x³ - 18x² - px - q.

so, (2x - 1) is a factor of 8x⁴ - 8x³ - 18x² - px - q.

so, 1/2 is a zero of given polynomial, 8x⁴ - 8x³ - 18x² - px - q..

so, 8(1/2)⁴ - 8(1/2)³ - 18(1/2)² - p(1/2) - q = 0

⇒1/2 - 1 - 9/2 - p/2 - q = 0

⇒-5 - p/2 - q = 0

⇒p + 2q + 10 = 0......(1)

now, 8x⁴ - 8x³ - 18x² - px - q

= 2x²(4x² - 4x + 1) - 20x² - px - q

= 2x² (4x² - 4x + 1) - 5(4x² + px/5 + q/5)

is 8x⁴ - 8x³ - 18x² - px - q exactly divisible by 4x² - 4x + 1 ?

then, 4x² + px/5 + q/5 = 4x² - 4x + 1

p = -20 and q = 5

now check equation (1), by putting p = -20 and q = 5.

i.e., -20 + 2 × 5 + 10 = 0

hence, p = -20 and q = 5

Answer 2

Step-by-step explanation:

8x⁴-8x³-18x-px-q divisible by 4x2-4x+1