Question

what are the zeros to the function?
f(x) = (x - 7)(x² + 64)
*type the numbers separated by a comma. no spaces.
enter answer here

Explanation:

Step1: Set function to zero

To find the zeros of the function \( f(x) = (x - 7)(x^2 + 64) \), we set \( f(x) = 0 \). So we have the equation \( (x - 7)(x^2 + 64)=0 \).

Step2: Solve for x using zero - product property

The zero - product property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) (or both).

• Case 1: When \( x - 7=0 \), we can solve for \( x \) by adding 7 to both sides of the equation. So \( x=7 \).
• Case 2: When \( x^2 + 64 = 0 \), we first subtract 64 from both sides of the equation to get \( x^2=- 64 \). Then we take the square root of both sides. Remember that \( \sqrt{-a}=\pm i\sqrt{a} \) for \( a>0 \). So \( x=\pm\sqrt{- 64}=\pm8i \). But since the problem seems to be asking for real - valued zeros (as there is no mention of complex numbers in the problem statement's context of "type the numbers"), we consider the real - valued zero from the first case and the complex zeros. However, if we assume that we are to include complex zeros, the zeros are \( 7,8i, - 8i \). But if we consider only real zeros, the zero is \( 7 \). But looking at the problem statement "type the numbers separated by a comma", and if we consider complex numbers as "numbers" (in the context of complex numbers), the zeros are \( 7,8i, - 8i \). But if we made a mistake and the function was \( (x - 7)(x^2-64) \), the zeros would be \( 7,8, - 8 \). But based on the given function \( f(x)=(x - 7)(x^2 + 64) \), the zeros are found as follows:

From \( x - 7 = 0\Rightarrow x = 7 \)

From \( x^2+64=0\Rightarrow x^2=-64\Rightarrow x=\pm\sqrt{-64}=\pm8i \)

Answer:

7,8i, - 8i

(If the problem intended to have \( x^2 - 64\) instead of \( x^2+64\), the answer would be 7,8, - 8. But based on the given function \( f(x)=(x - 7)(x^2 + 64) \), the zeros are 7, 8i, - 8i)