Question

1. the polynomial $p(x) = (1 - 3x)(x^2 + 2)$ is graphed in the coordinate plane. what is the end behavior for the function?

Explanation:

Step1: Expand the polynomial

First, we expand \( p(x) = (1 - 3x)(x^2 + 2) \). Using the distributive property (FOIL method for binomial and trinomial multiplication), we get:

$$
LATEXBLOCK0
$$

Step2: Determine the leading term

The leading term of a polynomial is the term with the highest degree. For the polynomial \( p(x)=-3x^3+x^2 - 6x + 2 \), the leading term is \( -3x^3 \). The degree of the polynomial is the exponent of the leading term, which is \( 3 \) (odd), and the leading coefficient is \( - 3 \) (negative).

Step3: Analyze end - behavior based on degree and leading coefficient

For a polynomial function \( y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \):

• If the degree \( n \) is odd:
• If the leading coefficient \( a_n>0 \), as \( x

ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \).

• If the leading coefficient \( a_n < 0 \), as \( x

ightarrow+\infty \), \( y
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow+\infty \).

Since our polynomial has an odd degree (\( n = 3 \)) and a negative leading coefficient (\( a_3=-3<0 \)):

• As \( x

ightarrow+\infty \), \( p(x)
ightarrow-\infty \)

• As \( x

ightarrow-\infty \), \( p(x)
ightarrow+\infty \)

Answer:

As \( x
ightarrow+\infty \), \( p(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( p(x)
ightarrow+\infty \)