f(x)=2x3−3px2+x+4pwhere p is an integer. Given that (x−4)is afactor off(x) .(a) Calculate the value of p . Using the value of p,(b) find the remainder when f(x)  is divided by (x+2)( c ) factorise f(x) completely

(a) Since (x-4) is a factor of f(x), we can use the Factor Theorem which states that if (x-a) is a factor of f(x), then f(a) = 0.

So, we substitute x = 4 into the equation:

f(4) = 2(4)^3 - 3p(4)^2 + 4 + 4p = 0 = 2(64) - 3p(16) + 4 + 4p = 0 = 128 - 48p + 4 + 4p = 0 = 132 - 44p = 0

Solving for p, we get:

44p = 132 p = 132 / 44 p = 3

(b) To find the remainder when f(x) is divided by (x+2), we substitute x = -2 into the equation:

f(-2) = 2(-2)^3 - 33(-2)^2 - 2 + 43 = 2(-8) - 334 - 2 + 12 = -16 - 36 - 2 + 12 = -42

So, the remainder is -42.

(c) To factorise f(x) completely, we already know that (x-4) is a factor. Now we need to find the other factors.

We can do this by dividing f(x) by (x-4). The quotient we get will be the other factor.

So, f(x) = (x-4)(2x^2 + 5x - 3)

We can further factorise the quadratic expression:

2x^2 + 5x - 3 = (2x - 1)(x + 3)

So, the complete factorisation of f(x) is:

f(x) = (x - 4)(2x - 1)(x + 3)