Question

sign chart from factored polynomial (level 2)
question
identify the real roots of the function below then use those (real) roots to fill in the sign table below.
$f(x) = -(x - 5)(3x - 1)(x^2 + 3)$
identify all real roots. use commas to separate.

Explanation:

Step1: Recall root definition

A root of a function \( f(x) \) is a value \( x \) such that \( f(x) = 0 \). For a factored polynomial, we set each factor equal to zero and solve for \( x \).

Step2: Analyze each factor

• For the factor \( -(x - 5) \): Set \( -(x - 5)=0 \), multiply both sides by -1: \( x - 5 = 0 \), so \( x = 5 \).
• For the factor \( (3x - 1) \): Set \( 3x - 1 = 0 \), add 1 to both sides: \( 3x = 1 \), divide by 3: \( x=\frac{1}{3} \).
• For the factor \( (x^2 + 3) \): Set \( x^2 + 3 = 0 \), then \( x^2=-3 \). Since the square of a real number cannot be negative, there are no real roots from this factor.

Answer:

\( \frac{1}{3}, 5 \)