Question

4 of 10
what are the zeros to the function?
f(x) = (2x + 10)(x² + 1)
*type the numbers separated by a comma. no spaces.
enter answer here

Explanation:

Step1: Set \( f(x) = 0 \)

To find the zeros of the function \( f(x)=(2x + 10)(x^{2}+1) \), we set \( f(x)=0 \). So we have the equation \((2x + 10)(x^{2}+1)=0\).

Step2: Solve for \( x \) from each factor

According to the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).

• For the factor \( 2x+10 = 0 \):

Subtract 10 from both sides: \( 2x=- 10 \).
Divide both sides by 2: \( x=-5 \).

• For the factor \( x^{2}+1 = 0 \):

Subtract 1 from both sides: \( x^{2}=-1 \).
Taking the square root of both sides, we know that in the set of real numbers, the square root of a negative number is not defined. But if we consider the complex number system, \( x=\pm i \) (where \( i=\sqrt{-1} \)). However, since the problem does not specify the number system and usually when we talk about zeros of a function in the context of a basic algebra problem (without specifying complex numbers), we consider real zeros first. But if we consider complex zeros, the zeros are \( - 5,i,-i \). But if we assume we are working in the real number system, the only real zero is \( x = - 5 \). But let's check the problem statement again. The problem says "type the numbers", if we consider complex numbers as "numbers" (in the broad sense), the zeros are \( - 5,i,-i \). But maybe the problem expects real zeros. Wait, let's re - examine the function. The function is \( f(x)=(2x + 10)(x^{2}+1) \).
If we solve \( 2x+10 = 0\), we get \( x=-5 \).
If we solve \( x^{2}+1=0\), \( x^{2}=-1\), so \( x = i \) or \( x=-i \) (where \( i=\sqrt{-1} \)).

Answer:

-5,i,-i