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 $f(x)$
\[
f(x)=4 x^{6}-64 x^{4}+x^{3}-15
\]
The remainder is
Answer
Final Answer: The remainder when \(f(x)\) is divided by \(x+4\) is \(\boxed{-79}\). \(x+4\) is not a factor of \(f(x)\).
Step 1 ：Given the polynomial function \(f(x)=4 x^{6}-64 x^{4}+x^{3}-15\), we are asked to find the remainder when \(f(x)\) is divided by \(x+4\) and determine whether \(x+4\) is a factor of \(f(x)\).
Step 2 ：We use the remainder theorem which states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x-a\) is equal to \(f(a)\). In this case, we are dividing by \(x+4\), so we need to evaluate \(f(-4)\) to find the remainder.
Step 3 ：Substitute \(x = -4\) into \(f(x)\), we get \(f(-4) = 4*(-4)^{6} - 64*(-4)^{4} + (-4)^{3} - 15 = -79\).
Step 4 ：The remainder when \(f(x)\) is divided by \(x+4\) is \(-79\).
Step 5 ：According to the factor theorem, if \(x+4\) is a factor of \(f(x)\), then \(f(-4)\) should be equal to 0. Since \(f(-4)\) is \(-79\), not 0, \(x+4\) is not a factor of \(f(x)\).
Step 6 ：Final Answer: The remainder when \(f(x)\) is divided by \(x+4\) is \(\boxed{-79}\). \(x+4\) is not a factor of \(f(x)\).
