Question

example: write the equation in factored form
you try it: write the equation in factored form

extension – sketching graphs of polynomials
sketch the graphs of each of the following polynomial functions:
a. $f(x) = x(x - 1)(x + 1)$
degree:
leading coefficient (+ or -):
zeros:
multiplicities:
y – intercept:
b. $f(x) = (x + 3)(x + 3)(x + 3)(x + 3)$
degree:
leading coefficient (+ or -):
zeros:
multiplicities:
y – intercept:

Explanation:

Part a: \( f(x) = x(x - 1)(x + 1) \)

Step 1: Find the Degree

The degree of a polynomial is the highest power of \( x \). When we expand \( x(x - 1)(x + 1) \), first multiply \( (x - 1)(x + 1)=x^{2}-1 \), then multiply by \( x \) to get \( x^{3}-x \). The highest power is 3.
Degree: \( 3 \)

Step 2: Leading Coefficient

The leading coefficient is the coefficient of the highest - degree term. In \( x^{3}-x \), the coefficient of \( x^{3} \) is 1 (positive).
Leading Coefficient: \( + \)

Step 3: Find the Zeros

To find the zeros, set \( f(x)=0 \), so \( x(x - 1)(x + 1)=0 \). Using the zero - product property, \( x = 0 \) or \( x-1=0\Rightarrow x = 1 \) or \( x + 1=0\Rightarrow x=-1 \).
Zeros: \( x = 0,x = 1,x=-1 \)

Step 4: Multiplicities

The multiplicity of a zero is the number of times its corresponding factor appears. For \( x = 0 \), the factor is \( x \) (appears 1 time), for \( x = 1 \), the factor is \( (x - 1) \) (appears 1 time), and for \( x=-1 \), the factor is \( (x + 1) \) (appears 1 time).
Multiplicities: \( 1,1,1 \)

Step 5: Find the Y - Intercept

To find the y - intercept, set \( x = 0 \) in \( f(x) \). \( f(0)=0\times(0 - 1)\times(0 + 1)=0 \).
Y - Intercept: \( 0 \)

Part b: \( f(x)=(x + 3)(x + 3)(x + 3)(x + 3)=(x + 3)^{4} \)

Step 1: Find the Degree

The degree of a polynomial in the form \( (x + a)^{n} \) is \( n \). Here \( n = 4 \), so the degree is 4.
Degree: \( 4 \)

Step 2: Leading Coefficient

When we expand \( (x + 3)^{4}=x^{4}+12x^{3}+54x^{2}+108x + 81 \), the coefficient of \( x^{4} \) is 1 (positive).
Leading Coefficient: \( + \)

Step 3: Find the Zeros

Set \( f(x)=0 \), so \( (x + 3)^{4}=0 \). Then \( x+3 = 0\Rightarrow x=-3 \).
Zeros: \( x=-3 \)

Step 4: Multiplicities

The factor \( (x + 3) \) appears 4 times, so the multiplicity of the zero \( x=-3 \) is 4.
Multiplicities: \( 4 \)

Step 5: Find the Y - Intercept

Set \( x = 0 \) in \( f(x) \). \( f(0)=(0 + 3)^{4}=81 \).
Y - Intercept: \( 81 \)

Answer:

s:

Part a:

Degree: \( \boldsymbol{3} \)
Leading Coefficient: \( \boldsymbol{+} \)
Zeros: \( \boldsymbol{x = 0,x = 1,x=-1} \)
Multiplicities: \( \boldsymbol{1,1,1} \)
Y - Intercept: \( \boldsymbol{0} \)

Part b:

Degree: \( \boldsymbol{4} \)
Leading Coefficient: \( \boldsymbol{+} \)
Zeros: \( \boldsymbol{x=-3} \)
Multiplicities: \( \boldsymbol{4} \)
Y - Intercept: \( \boldsymbol{81} \)