Question

given the function (p(x) = x^3 + x^2 - 20x).

the (y)-intercept is

the (x)-intercepts is/are

when (x
ightarrow infty), (y
ightarrow) ?

when (x
ightarrow -infty), (y
ightarrow) ?

Explanation:

⚡ Using what you learned: Understanding Polynomial Functions

Step 1: Find the \(y\)-intercept

To find the \(y\)-intercept, evaluate the function at \(x = 0\):

$$ P(0) = 0^3 + 0^2 - 20(0) = 0 $$

The \(y\)-intercept is \(0\) (or written as an ordered pair, \((0, 0)\)).

Step 2: Find the \(x\)-intercepts

To find the \(x\)-intercepts, set \(P(x) = 0\) and solve for \(x\):

$$ x^3 + x^2 - 20x = 0 $$

Factor out the greatest common factor, \(x\):

$$ x(x^2 + x - 20) = 0 $$

Factor the quadratic trinomial:

$$ x(x + 5)(x - 4) = 0 $$

Set each factor to zero:

$$ x = 0, \quad x = -5, \quad x = 4 $$

The \(x\)-intercepts are \(0, -5, 4\) (or written as ordered pairs, \((0,0), (-5,0), (4,0)\)).

Step 3: Determine the end behavior

The leading term of the polynomial \(P(x) = x^3 + x^2 - 20x\) is \(x^3\).

Since the degree is odd (\(3\)) and the leading coefficient is positive (\(1\)):

• As \(x

ightarrow \infty\), \(y
ightarrow \infty\)

• As \(x

ightarrow -\infty\), \(y
ightarrow -\infty\)

Answer:

• The \(y\)-intercept is \(0\) (or \((0,0)\))
• The \(x\)-intercepts is/are \(0, -5, 4\) (or \((0,0), (-5,0), (4,0)\))
• **When \(x

ightarrow \infty\), \(y
ightarrow\)** \(\infty\)

• **When \(x

ightarrow -\infty\), \(y
ightarrow\)** \(-\infty\)