Use the Remainder Theorem to find the remainder for (4x^3+4x^2+2x+3)/(x-1) and state whether or not the binomial is a factor of the polynomial.

The remainder theorem states that when a polynomial is divided by a binomial, the value that makes the binomial equal 0 can be substituted into the polynomial to obtain the remainder.

In other words, we can plug in 1 into the polynomial, as (1 - 1) = 0

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

4 + 4 + 2 + 3 = 13

As the polynomial equals 13, the remainder when the polynomial is divided is 13/(x -1), so it’s NOT a factor of the polynomial.

If the remainder were 0, then (x - 1) WOULD be a factor of the polynomial, but it’s not.

To sum it up, the remainder is 13/(x - 1), and the binomial is NOT a factor of the polynomial.

Iwould say they are complementary because that’s the division of a 90 degree angle and it’s split between the middle of them