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Wize High School Grade 12 Pre-Calculus Textbook > 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)
where a∈Ra\in\mathbb{R} , n∈Wn\in\mathbb{W}, and x1, x2,..., xnx_1,~x_2,...,~x_n are the roots (i.e. xx-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 (xx-intercept) at some value x=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)^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)^n, where n is even, then the graph of the function will touch the root at x1 x_1~ and then bounce off, heading in the other direction.
y=(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, where n is odd, then the graph of the function will pass through the root at x1x_1
y=(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).

Determine:
  1. The degree of the function
  2. The xx-intercepts & their multiplicity
  3. The yy-intercept
  4. Rough sketch

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. xx-intercepts & multiplicity:
  • x = 5 with multiplicity of 1
  • x = 1 with multiplicity of 2
  • x = -2 with multiplicity of 3
3. yy-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}

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)^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
  • xx-Intercepts:
  • −1-1 with a multiplicity of 1
  • 22 with a multiplicity of 2
  • 44 with a multiplicity of 5
  • As x→∞, y→∞x→−∞, y→∞x\rightarrow\infin,~y\rightarrow\infin\newline x\rightarrow-\infin,~y\rightarrow\infty

Sketch of the function:


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Example: Factoring & Graphing Polynomials

Find a function f(x)f(x) with the following properties:
  • xx-intercepts at:
  • −3-3 with a multiplicity of 2
  • 11 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)^2.

FInd 'aa' 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}

Therefore, f(x)=15(x−1)3(x+3)2f(x)=\dfrac{1}{5}(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+e. What must be true about f(x) f(x)~ if it is a negative polynomial function?

Select all that apply.

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)

Practice: Factoring & Graphing Polynomials

A function f(x)f(x) has the following properties:
  • xx-intercepts at:
  • −2-2 with a multiplicity of 2
  • 1.51.5 with a multiplicity of 3
  • 22 with a multiplicity of 4
  • yy-intercept at (0, 10)
True or False: The point (−1,3)(-1, 3) is on f(x)f(x)?
Extra Practice
Factoring & Graphing Polynomials
Practice: Factoring & Graphing Polynomials
A function f(x)f(x) has the following properties:
xx-intercepts at:
Factoring & Graphing Polynomials
Practice: Factoring & Graphing Polynomials
A function f(x)f(x) has the following properties:
xx-intercepts at:
Factoring & Graphing Polynomials
Practice: Factoring & Graphing Polynomials
Sketch 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)
Factoring & Graphing Polynomials
Practice: Factoring & Graphing Polynomials
Suppose f(x)=ax3+bx2+cx+df(x)=ax^3+bx^2+cx+d. What must be true about 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+d. What must be true about f(x) f(x)~ if it is a positive polynomial function?
Select all that apply.