Wize High School Algebra II Textbook (Common Core) > Polynomial Functions

Factoring & Graphing Polynomials

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Factoring & Graphing Polynomials
Polynomial functions can be expressed in factored form,
f(x)=a(x−x1)(x−x2)...(x−xn)f(x)=a(x-x_1)(x-x_2)...(x-x_n)f(x)=a(x−x1​)(x−x2​)...(x−xn​)
where a∈Ra\in\mathbb{R}a∈R , n∈Wn\in\mathbb{W}n∈W, and x1, x2,..., xnx_1,~x_2,...,~x_nx1​, x2​,..., xn​ are the roots (i.e. xxx-intercepts)

Multiplicity of Roots
The multiplicity of roots is the number of occurrences of a root in the complete factorization of a polynomial.

A polynomial can have more than one root (xxx-intercept) at some value x=ax=ax=a.

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Example 1

Let's look at the graph of the function f(x)=(x+1)3(x−3)2f(x)=(x+1)^3(x-3)^2f(x)=(x+1)3(x−3)2
Root at -1 with a multiplicity of 3
Root at 3 with a multiplicity of 2


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Even Multiplicity of Roots
If f(x)=(x−x1)nf(x)=(x-x_1)^nf(x)=(x−x1​)n, where n is even, then the graph of the function will touch the root at x1 x_1~x1​  and then bounce off, heading in the other direction.
  
y=(x+1)2(x−2)4y=(x+1)^2(x-2)^4y=(x+1)2(x−2)4

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Odd Multiplicity of Roots
If f(x)=(x−x1)n,f(x)=(x-x_1)^n,f(x)=(x−x1​)n, where n is odd, then the graph of the function will pass through the root at x1x_1x1​
   
y=(x+1)3(x−2)5y=(x+1)^3(x-2)^5y=(x+1)3(x−2)5

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Example: Factoring & Graphing Polynomials
Let f(x)=2(x−1)2(x+2)3(x−5)f(x)=2(x-1)^2(x+2)^3(x-5)f(x)=2(x−1)2(x+2)3(x−5). 

Determine:
The degree of the function 
The xxx-intercepts & their multiplicity
The yyy-intercept
Rough sketch

f(x)=2(x−1)2(x+2)3(x−5)f(x)=2(x-1)^2(x+2)^3(x-5)f(x)=2(x−1)2(x+2)3(x−5)

1. Degree: 6 
2. xxx-intercepts & multiplicity:
x = 5 with multiplicity of 1
x = 1 with multiplicity of 2
x = -2 with multiplicity of 3
3. yyy-intercept:

f(0)=2(0−1)2(0+2)3(0−5)=−80\begin{array}{rcl}
f(0)&=&2(0-1)^2(0+2)^3(0-5)\\
&=&-80
\end{array}f(0)​==​2(0−1)2(0+2)3(0−5)−80​

4. Sketch:

Example: Factoring & Graphing Polynomials
Graph f(x)=(x+1)(x−2)2(x−4)5f(x)=(x+1)(x-2)^2(x-4)^5f(x)=(x+1)(x−2)2(x−4)5, stating the following properties:
Positive/Negative & Even/Odd Polynomial
Symmetry
X-Intercepts & Multiplicity
End Behaviour

The function f(x) has the following properties:
Positive, Even Polynomial
No Symmetry
xxx-Intercepts:
−1-1−1 with a multiplicity of 1
222 with a multiplicity of 2
444 with a multiplicity of 5
As x→∞, y→∞x→−∞, y→∞x\rightarrow\infin,~y\rightarrow\infin\newline
x\rightarrow-\infin,~y\rightarrow\inftyx→∞, y→∞x→−∞, y→∞

Sketch of the function:

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Example: Factoring & Graphing Polynomials
Find a function f(x)f(x)f(x) with the following properties:
xxx-intercepts at:
−3-3−3 with a multiplicity of 2
111 with a multiplicity of 3
Passes through the point (2, 5)

Let f(x)=a(x−1)3(x+3)2f(x)=a(x-1)^3(x+3)^2f(x)=a(x−1)3(x+3)2. 

FInd 'aaa' using (2, 5):
5=a(2−1)3(2+3)25=25a∴a=155=a(2-1)^3(2+3)^2\newline{}\newline
5=25a\newline{}\newline
\therefore a=\frac{1}{5}5=a(2−1)3(2+3)25=25a∴a=51​

Therefore, f(x)=15(x−1)3(x+3)2f(x)=\dfrac{1}{5}(x-1)^3(x+3)^2f(x)=51​(x−1)3(x+3)2

Practice: Factoring & Graphing Polynomials
Suppose f(x)=ax4+bx3+cx2+dx+ef(x)=ax^4+bx^3+cx^2+dx+ef(x)=ax4+bx3+cx2+dx+e. What must be true about f(x) f(x)~f(x)   if it is a negative polynomial function?

Select all that apply.

A) a<0a<0a<0

B) a>0a>0a>0

C) f(x)f(x)f(x) begins in quadrant 1 and ends in quadrant 2

D) f(x)f(x)f(x) begins in quadrant 3 and ends in quadrant 4

E) Minimum number of x-intercepts is 1

F) Minimum number of x-intercepts is 0

I don't know

Practice: Factoring & Graphing Polynomials

Sketch f(x)=150(x+5)2(x+2)4(2x−1)f(x)=\frac{1}{50}(x+5)^2(x+2)^4(2x-1)f(x)=501​(x+5)2(x+2)4(2x−1)

Practice: Factoring & Graphing Polynomials
A function f(x)f(x)f(x) has the following properties:
xxx-intercepts at:
−2-2−2 with a multiplicity of 2
1.51.51.5 with a multiplicity of 3
222 with a multiplicity of 4
yyy-intercept at (0, 10)
True or False: The point (−1,3)(-1, 3)(−1,3) is on f(x)f(x)f(x)?

Answer

I don't know

Extra Practice

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
A function f(x)f(x)f(x) has the following properties:
xxx-intercepts at:

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
A function f(x)f(x)f(x) has the following properties:
xxx-intercepts at:

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
Sketch f(x)=−(2x−1)3(x−6)f(x)=-(2x-1)^3(x-6)f(x)=−(2x−1)3(x−6)

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
Sketch f(x)=125(x+2)2(x−1)f(x)=\frac{1}{25}(x+2)^2(x-1)f(x)=251​(x+2)2(x−1)

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
Suppose f(x)=ax3+bx2+cx+df(x)=ax^3+bx^2+cx+df(x)=ax3+bx2+cx+d. What must be true about f(x) f(x)~f(x)   if it is a negative polynomial function?
Select all that apply.

Factoring & Graphing Polynomials

Practice: Factoring & Graphing Polynomials
Suppose f(x)=ax6+bx3+cx+df(x)=ax^6+bx^3+cx+df(x)=ax6+bx3+cx+d. What must be true about f(x) f(x)~f(x)   if it is a positive polynomial function?
Select all that apply.