Question

sketch a graph of what each function may look like an

1. ( y = (x - 2)(x + 4)^2(x - 5) )

degree:____ type:_______
end behavior:
as ( x \to infty )
( f(x) \to )______
as ( x \to -infty )
( f(x) \to )______
zero:____ mult:____
zero:____ mult:____
zero:____ mult:____
graph with x-axis from -8 to 8 and y-axis from -8 to 8, grid lines

Explanation:

Step1: Find the degree

To find the degree of the polynomial \( y=(x - 2)(x + 4)^2(x - 5) \), we add the exponents of each linear factor. The exponents are 1 (for \( x - 2 \)), 2 (for \( (x + 4)^2 \)), and 1 (for \( x - 5 \)). So, \( 1+2 + 1=4 \). So the degree is 4.

Step2: Determine the type

Since the degree is 4 (an even number), the polynomial is a quartic (or 4th - degree) polynomial. And since the leading coefficient (when we expand the polynomial, the coefficient of \( x^4 \) will be 1, which is positive) is positive, it is a quartic (even - degree) polynomial.

Step3: Analyze end behavior

For a polynomial \( y = a_nx^n+\cdots+a_0 \), if \( n \) is even:

• If \( a_n>0 \), as \( x

ightarrow\infty \), \( y
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow\infty \).
Here, the leading coefficient (the coefficient of the highest - degree term) is positive (when we expand \( (x - 2)(x + 4)^2(x - 5) \), the leading term comes from multiplying the leading terms of each factor: \( x\times x^2\times x=x^4 \), so the leading coefficient is 1>0) and the degree \( n = 4 \) (even). So as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \).

Step4: Find the zeros and their multiplicities

• For the factor \( (x - 2) \), set \( x - 2 = 0\), we get \( x = 2 \). The multiplicity is 1 (since the exponent of \( (x - 2) \) is 1).
• For the factor \( (x + 4)^2 \), set \( x+4 = 0\), we get \( x=-4 \). The multiplicity is 2 (since the exponent of \( (x + 4) \) is 2).
• For the factor \( (x - 5) \), set \( x - 5 = 0\), we get \( x = 5 \). The multiplicity is 1 (since the exponent of \( (x - 5) \) is 1).

Answer:

Degree: \( \boldsymbol{4} \)
Type: \( \boldsymbol{\text{Quartic (4th - degree) Polynomial}} \)
End Behavior:
as \( x
ightarrow\infty \), \( f(x)
ightarrow\boldsymbol{\infty} \)
as \( x
ightarrow-\infty \), \( f(x)
ightarrow\boldsymbol{\infty} \)
Zeros and Multiplicities:
Zero: \( \boldsymbol{2} \), Mult: \( \boldsymbol{1} \)
Zero: \( \boldsymbol{-4} \), Mult: \( \boldsymbol{2} \)
Zero: \( \boldsymbol{5} \), Mult: \( \boldsymbol{1} \)