Question

havior of the polynomial function $f(x) = 8x^7 - 4x^4 + 5x^3 - 2x + 6$?

Explanation:

Step1: Identify Degree and Leading Coefficient

The polynomial \( f(x) = 8x^7 - 4x^4 + 5x^3 - 2x + 6 \) has a degree of 7 (the highest power of \( x \)) and a leading coefficient of 8 (the coefficient of the term with the highest degree).

Step2: Apply End - Behavior Rules

For a polynomial function \( f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), the end - behavior is determined by the degree \( n \) and the leading coefficient \( a_n \).

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

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

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

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

Since the degree \( n = 7 \) (odd) and the leading coefficient \( a_7=8>0 \), we have:
As \( x
ightarrow+\infty \), \( f(x)
ightarrow+\infty \) (because the leading term \( 8x^7 \) will dominate and when \( x \) is a large positive number, \( 8x^7 \) is a large positive number).
As \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \) (because when \( x \) is a large negative number, \( x^7 \) is negative (since the exponent is odd) and multiplying by 8 (a positive number) gives a large negative number, and the other terms become negligible in comparison to \( 8x^7 \) as \( |x| \) becomes very large).

Answer:

As \( x
ightarrow+\infty \), \( f(x)
ightarrow+\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)