Question

determine whether (x+1) of the polynomial x^4+4x^3+2x^2-2x+1 or not

Answer 1

In this question we need to see that whether (x + 1) is a factor of the given polynomial or not.

First of all, you need to find the zero of (x + 1)

For finding zeroes of polynomials, we use the polynomial equation.

so, if x + 1 = 0

=) x = -1

So, the zero of (x + 1) is -1.

Now take, f(x) = x^4 + 4x^3 + 2x^2 - 2x + 1

We know from the Factor Theorem that when the remainder is 0, the divisor (x - a) will be a factor of f(x).

So, if in this case, the remainder is 0, the divisor will be a factor of f(x).

For that, we need to find f(-1), which is:

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

=) 1 - 4 + 2 + 2 + 1

=) -3 + 2 + 2 + 1

=) -1 + 2 + 1

=)  2

So, the remainder is 2.

Clearly, (x + 1) is not a factor of f(x).

Hope it helps.