Please help!! I do not understand any of this!!
Use the remainder theorem to find the remainder when f(x) is divided by x+4. Then use the factor theorem to determine whether x+4 is a factor of f(x). f(x)=4x^6-64x^4+x^2-18
Sure, I'd be happy to help you understand this problem!
To begin, let's find the remainder when f(x) is divided by x+4 using the remainder theorem. The remainder theorem states that if we divide a polynomial f(x) by x-a, then the remainder is equal to f(a).
In this case, we want to divide f(x) = 4x^6 - 64x^4 + x^2 - 18 by x+4. To use the remainder theorem, we substitute x = -4 into f(x) and evaluate the expression.
Let's calculate this step by step:
1. Replace x in f(x) with -4:
f(-4) = 4(-4)^6 - 64(-4)^4 + (-4)^2 - 18
2. Simplify the expression:
f(-4) = 4(4096) - 64(256) + 16 - 18
f(-4) = 16384 - 16384 + 16 - 18
f(-4) = 0
Therefore, the remainder when f(x) is divided by x+4 is 0.
Now, let's use the factor theorem to determine whether x+4 is a factor of f(x). The factor theorem states that if a polynomial f(x) has a factor x-a, then f(a) = 0.
In our case, since the remainder we calculated using the remainder theorem is 0, we can conclude that x+4 is indeed a factor of f(x).
I hope this explanation clarifies the problem for you. Let me know if you have any further questions!