Question

test review #3, chapter 5
score: 6.5/37 answered: 6/37
question 7
given the function ( p(x) = x^3 + 3x^2 - 10x ).
the ( y )-intercept is
the ( x )-intercepts is/are
when ( x \to infty ), ( y \to )?
when ( x \to -infty ), ( y \to )?
question help: video

Explanation:

Step1: Find y-intercept (set $x=0$)

$P(0) = 0^3 + 3(0)^2 - 10(0) = 0$

Step2: Factor for x-intercepts (set $P(x)=0$)

$x^3 + 3x^2 - 10x = x(x^2 + 3x - 10) = x(x+5)(x-2) = 0$
Solve for $x$: $x=0$, $x=-5$, $x=2$

Step3: Analyze end behavior (leading term $x^3$)

As $x \to \infty$, $x^3 \to \infty$, so $y \to \infty$

Step4: Analyze end behavior (negative x)

As $x \to -\infty$, $x^3 \to -\infty$, so $y \to -\infty$

Answer:

The y-intercept is $0$
The x-intercepts are $-5$, $0$, $2$
When $x \to \infty$, $y \to \infty$
When $x \to -\infty$, $y \to -\infty$