Factor and cubes

Question

find a so that x + 1 is a factor

4x^5 + 3x^2 + 2x + a

Answer 1

find a so that x + 1 is a factor

4x^5 + 3x^2 + 2x + a

If x+1 is a factor, then when x= -1, the quintic is zero;

f(x) = 4x^5 + 3x^2 + 2x + a

f(-1) = 0

4(-1)^5 + 3(-1)^2 + 2(-1) + a = 0

-4 + 3 - 2 + a = 0

a = -3

Answer 2

Since (x+1) is given as a factor, then synthetic division by -1 must result in a remainder of zero.

-1 4 0 0 3 2 a

-4 4 -4 1 -3

---------------------------

4 -4 4 -1 3 0

So a must be +3.

Multiply (x + 1) ( 4x⁴ -4x³ + 4x² - x + 3) to confirm.