Question

question
look at your notes from yesterday
what are the zeros of $f(x) = (5x - 2)(x + 6)^2$

Explanation:

Step1: Set the function equal to zero

To find the zeros of the function \( f(x)=(5x - 2)(x + 6)^{2} \), we set \( f(x)=0 \). So we have the equation:
\[
(5x - 2)(x + 6)^{2}=0
\]

Step2: Apply the zero - product property

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

• Case 1: Set \( 5x-2 = 0 \)

Solve for \( x \):
Add 2 to both sides of the equation: \( 5x=2 \)
Divide both sides by 5: \( x=\frac{2}{5} \)

• Case 2: Set \( (x + 6)^{2}=0 \)

Take the square root of both sides. Since \( y^{2}=0 \) implies \( y = 0 \), we have \( x+6 = 0 \)
Solve for \( x \): Subtract 6 from both sides, so \( x=-6 \) (this root has a multiplicity of 2, but we still list it as a zero of the function)

Answer:

The zeros of the function \( f(x)=(5x - 2)(x + 6)^{2} \) are \( x=\frac{2}{5} \) and \( x=-6 \) (with \( x = - 6 \) having a multiplicity of 2).