Question

X-3 and 2x-1 are factors of 2x^3-px^2-2qx+q. Find the values of p and q.

Answer 1

p =1

q = 9

Explanation:

f(x) = 2x³ - px² + 2qx + q

(x - 3) is a factor of f(x)

⇒f(3) = 0

2(3)³ - p*3² - 2q*3 +q = 0

2*27 - 9p - 6q + q = 0

54 - 9p - 5q = 0

-9p - 5q = -54 -------------------(I)

(2x - 1) is a factor of f(x)

2x - 1 = 0

2x = 1

`\st x = (1)/(2)`

f(1/2) = 0

`2*((1)/(2))^(3)-p*((1)/(2))^(2)-2q*(1)/(2)+q=0\\\\2*(1)/(8)-p*(1)/(4)-q+q = 0\\\\(1)/(4)-(1)/(4)p =0\\\\[Multiply the entire equation by 4]\\\\4*(1)/(4)-4*(1)/(4)p=0\\\\`

1 - p = 0

-p = -1

p = 1

Substitute p =1 in equation (I)

-9*1 - 5q = -54

-9 - 5q = -54

-5q = -54 + 9

-5q = -45

q = -45/(-5)

q = 9