Question

Find the constant c such that the denominator will divide evenly into the numerator
x^5-2X^2+x+c/ X+2

Answer 1

When a polynomial P(x) is divided by x-b, the remainder is P(b)

So we have our polynomial P(x)=x^5-2x^2+x+c

and we are dividing it by x+2. If we look at x-b we can see that if b is -2 then: x-(-2) = x+2

so now we know b and we know our P(x). They tell us that they want it to divide evenly in, so the remained will be 0, which means P(b)=0

so we just to to set up the equation now and then we can solve for c

Answer 2

You could also do this by synthetic division if you don't know what I was talking about before

Answer 3

There's another way to do it as well.

x^5 + 2x^4 - 2x^4 - 4x^3 + 4x^3 + 8x^2 - 10x^2 -20x +21x +42
= x^4(x + 2) -2x^3(x + 2) + 4x^2(x + 2) - 10x(x + 2) + 21(x + 2)
= (x + 2)(x^4 - 2x^3 + 4x^2 - 10x + 21)

Anyway since we were able to factor out x + 2, you realize that if c = 42, then x + 2 divides the numerator easily.