use the remainder theorem to find the remainder when f(x) is divided by x+2. then use the factor theorem to determine whether x+2 is a factor of f(x).
f(x)=4x^6+36x^3+32
please show work
To find the remainder when f(x) is divided by x+2, we can use the Remainder Theorem. According to the theorem, if you divide a polynomial f(x) by x-a, where a is a constant, and you find the remainder R, then f(a) = R.
1. Step 1: Substitute x with -2 in f(x)
f(-2) = 4(-2)^6 + 36(-2)^3 + 32
Evaluate each term of the expression:
f(-2) = 4(64) + 36(-8) + 32
= 256 - 288 + 32
= 0
This means that the remainder when f(x) is divided by x+2 is zero.
Therefore, R = 0.
2. Step 2: Determine if x+2 is a factor using the Factor Theorem.
According to the Factor Theorem, if the remainder is zero when dividing a polynomial by x-a, then (x-a) is a factor of the polynomial.
In this case, since the remainder was found to be zero, we can conclude that x+2 is indeed a factor of f(x).