(a)
Since f(x) has remainder 45 when divided by (x +1) we have f(-1) = 45. That is:
f(-1) = 6(-1)^3 + 3(-1)^2 - A + B = -6 + 3 - A + B = 45
So:
B - A - 3 = 45
Giving:
B - A = 48
(b)
Since (2x + 1) is a factor of f(x) we have \displaystyle f\left(-\frac 1 2\right) = 0, so:
\displaystyle 6 \left(-\frac 1 2\right)^3 + 3 \left(-\frac 1 2\right)^2 - \frac 1 2 A + B = 0
So:
\displaystyle B = \frac 1 2 A
Then:
\displaystyle -\frac 1 2 A = 48
So A = -96 and B = -48.
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We know that f(x) will have the form:
(2x + 1)(3x^2 + ax + b) = 6x^3 + 2ax^2 + 3x^2 + 2bx + ax + b
Comparing coefficients have (2a + 3) = 3, so a = 0, and b = -48, so:
f(x) = (2x + 1)(3x^2 - 48) = 3(2x + 1)(x^2 - 16) = 3(2x+1)(x - 4)(x+4)
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