Question

question
how many x-intercepts would the function shown below have?
$f(x) = (x^2 - 5)(x + 5)$

Explanation:

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

To find the x - intercepts, we set the function equal to zero: \( (x^{2}-5)(x + 5)=0 \)

Step2: Solve each factor

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

• For the factor \( x + 5=0 \), we solve for \( x \) and get \( x=-5 \).
• For the factor \( x^{2}-5 = 0 \), we can rewrite it as \( x^{2}=5 \). Then, using the square - root property \( x=\pm\sqrt{5} \) (since if \( x^{2}=a,a\geq0 \), then \( x = \pm\sqrt{a} \)).

Step3: Count the number of solutions

The solutions to the equation \( f(x)=0 \) are \( x=-5,x = \sqrt{5},x=-\sqrt{5} \). Each of these solutions corresponds to an x - intercept. So we have 3 distinct solutions.

Answer:

The function \( f(x)=(x^{2}-5)(x + 5) \) has 3 x - intercepts.