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.
$\frac{1}{3},5$
click an area to change its sign.
answer attempt 2 out of 2
submit answer

Explanation:

Step1: Find real roots

Set each factor to zero: $x-5=0 \Rightarrow x=5$; $3x-1=0 \Rightarrow x=\frac{1}{3}$; $x^2+3=0$ has no real roots.

Step2: Determine sign intervals

Test $x<\frac{1}{3}$ (e.g., $x=0$): $-(0-5)(0-1)(0+3)=-( -5)(-1)(3)=-15$ (negative).
Test $\frac{1}{3}Test $x>5$ (e.g., $x=6$): $-(6-5)(18-1)(36+3)=-(1)(17)(39)=-663$ (negative).

Step3: Correct sign chart

Intervals: $(-\infty,\frac{1}{3})$: -, $(\frac{1}{3},5)$: +, $(5,\infty)$: -.

Answer:

$\frac{1}{3}, 5$