Question

set
topic: reviewing features of polynomials
some information has been given for each polynomial. fill in the missing information.
10.
function: ( f(x) = x^3 )
function in factored form:
end behavior:
as ( x \to -infty ), ( f(x) \to )
as ( x \to infty ), ( f(x) \to )
roots (with multiplicity):
degree:
value of leading co - efficient:
graph:
(grid graph here)
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74

Explanation:

Step1: Analyze Factored Form

The function is \( f(x) = x^3 \), which can be written as \( f(x)=x\cdot x\cdot x \) (factored form: three linear factors of \( x \)).

Step2: Determine Degree

The degree of a polynomial is the highest power of \( x \). For \( f(x)=x^3 \), the degree is \( 3 \).

Step3: Find Roots (with Multiplicity)

To find roots, set \( f(x) = 0 \), so \( x^3=0 \) gives \( x = 0 \) (with multiplicity \( 3 \), since the factor \( x \) appears three times).

Step4: Analyze End Behavior

For a polynomial \( f(x)=a_nx^n+\dots+a_0 \), end behavior is determined by the leading term \( a_nx^n \). Here, \( a_n = 1 \) (leading coefficient) and \( n = 3 \) (odd).

• As \( x

ightarrow -\infty \), for odd degree with positive leading coefficient, \( x^n
ightarrow -\infty \) (since \( (-\infty)^3=-\infty \)), so \( f(x)
ightarrow -\infty \).

• As \( x

ightarrow \infty \), \( x^3
ightarrow \infty \), so \( f(x)
ightarrow \infty \).

Step5: Leading Coefficient

The leading term is \( x^3 \), so the leading coefficient is \( 1 \).

Answer:

• Function in factored form: \( f(x)=x\cdot x\cdot x \) (or \( f(x)=x^3 \) as factored form with repeated factor)
• End behavior: As \( x

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

• Degree: \( 3 \)
• Roots (with multiplicity): \( x = 0 \) (multiplicity \( 3 \))
• Value of leading coefficient: \( 1 \)