Download Unit 3B Notes: Graphs of Polynomial Functions

Algebra 2A
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Unit 3B Notes: Graphs of Polynomial Functions
Topic 4: Factoring Polynomials (3.4)
Factors of polynomials have a remainder of _______ when they are divided into the polynomial.
Example: Determine whether the given binomial is a factor of the polynomial.
(x – 3); P(x) = x2 + 2x – 3
(x + 4); P(x) = 2x4 + 8x3 + 2x + 8
Strategies for Factoring Polynomials:
a. GCF
b. Grouping: Use with a polynomial with 4 terms.
Example 1: Factor: x3 – x2 – 25x + 25.
Step One Group terms with common factors.
Step Two Factor the common monomial.
Step Three Re-write as two binomial factors.
Step Four Factor the remaining polynomials using other techniques.
Example 2: Factor: 3x3 – 6x2 – 27x + 54.
Step One Group terms with common factors.
Step Two Factor the common monomial.
Step Three Re-write as two binomial factors.
Step Four Factor the remaining polynomial using other techniques.
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c. Sum and Difference of Two Cubes: Use with a binomial where both terms are cubes.
Example 1: Factor 125x3 – 8
Step One Identify “a” and “b”.
Step Two Apply the Sum/Difference Rule
Example 2: Factor 4x4 + 108x
Step Zero Factor the GCF
Step One Identify “a” and “b”.
Step Two Apply the Sum/Difference Rule
Topic 5: Finding Roots (aka. Zeroes, Solutions, x-Intercepts) of Polynomials (3.5 & 3.6)
I.
Solve by Factoring: Factor the polynomial and set each factor equal to _______. Solve!
Example: 4x6 + 4x5 – 24x4 = 0
Multiplicity: ______________________________________________________________________
Example 1: Solve by factoring x3 + 3x2 – 9x – 27 = 0
Example 2: Solve by factoring 5x4 + 15x3 + 15x2 + 5x = 0
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How to See Multiplicity on Graphs:
Even Multiplicity: ____________________________________________________
Odd Multiplicity: ____________________________________________________
II.
Rational Root Theorem:
Every _____________ rational factor of a polynomial can be found by finding the values of __________
p are factors of the constant term
q are factors of the leading coefficient
To determine which of these possible factors are zeros, we have to test the possible rational roots (no calc) or
we can graph the polynomial and use the trace/calc feature (calc)
Example 1 – NO CALC: Identify all the real roots of x3 + 3x2 – 4x – 12 = 0.
Step 1 Use the Rational Root Theorem to identify possible rational roots.
Step 2 Test the possible rational roots to find which are zeros.
Step 3 Factor the polynomial.
Example 2 – NO CALC: Identify all the real roots of 2x3 – 9x2 + 2 = 0.
Step 1 Use the Rational Root Theorem to identify possible rational roots.
Step 2 Test the possible rational roots to find which are zeros.
Step 3 Factor the polynomial using synthetic division.
Step 4 Find the remaining roots using the quadratic formula.
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Example 3 - CALC: Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.
Step 1 Use the Rational Root Theorem to identify possible rational roots.
Step 2 Graph y = 2x3 – 3x2 –10x – 4 to find the x-intercepts (use TRACE or CALC 2:ZERO)
Step 3 Factor the polynomial using synthetic division.
Step 4 Find the remaining roots using the quadratic formula.
III.
Writing Equations from Real Zeros: Like with quadratic equations - write the zeros as factors
and multiply. Simplify to standard form.
Example: Write a polynomial function with zeroes –3, 1, and 4.
IV.
Fundamental Theorem of Algebra:
Every polynomial of degree _________ has at least one _____, where a zero may be a ____________________.
The total number of real and imaginary zeroes is always ____________________________
Example 1 – NO CALC: Solve x4 – x3 + 2x2 – 4x – 8 = 0 by finding all roots.
Step 1 Use the Rational Root Theorem to identify the possible rational roots.
Step 2 Test the possible rational roots to find which are zeros.
Step 3 Factor the polynomial using synthetic division.
Step 4 Find the remaining roots.
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Example 2 – CALC: Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
Step 1 Use the Rational Root Theorem to identify the possible rational roots.
Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.
Step 3 Factor the polynomial using synthetic division.
Step 4 Find the remaining roots using the quadratic formula.
V.
Writing Equations from Real and Complex Zeroes
Remember: Square Roots and Imaginary Numbers must always come with their conjugate pair.
Example: Write the polynomial with zeros at 2 + i and 1.
Step 1 Identify all roots.
Step 2 Write the equation in factored form.
Step 3 Multiply.
Example: Write the polynomial with zeros
Step 1 Identify all roots.
and 2.
Step 2 Write the equation in factored form.
Step 3 Multiply.
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Topic 6: Describing Graphs of Polynomials (3.7)
I.
End Behavior: What happens as x approaches positive infinity and negative infinity.
a. Right = __________________
Notation: (x  ∞)
b. Left = ___________________
Notation: (x  –∞)
There is a pattern for end behavior based on the ____________ and ___________ of the leading coefficient.
Example 1: Q(x) = –x4 + 6x3 – x + 9
Leading Coefficient: _____
Degree: ______
As x  –∞, __________
As x  +∞, __________
Example 2: P(x) = 2x5 + 6x4 – x + 4
Leading Coefficient: _____
Degree: ______
End Behavior:
As x  –∞, __________
As x  +∞, __________
II.
Local Maxima & Minima
a. Turning Point: _____________________________________________________________
b. Local Maxima & Minima: ____________________________________________________
___________________________________________________________________________
*We can find local maxima and minima using the graphing calculators*
2ND CALC (above TRACE)  and 3:minimum or 4:maximum
Example: Graph f(x) = 2x3 – 18x + 1 on a calculator and find the values of the local maxima and minima.
III.
Sketching Graphs of Polynomial Functions
a. Points to include on your graphs:
i. The real zeros and y-intercept of the function
ii. Local Maxima and Minima
iii. End behavior (arrows)
Example: Sketch a graph of the function P(x) = x3 + 4x2 + x – 6.
Real Zeros: ____________
Local Maxima and Minima: ____________
End Behavior: ______________________
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Topic 7: Transformations of Polynomial Functions (3.8)
We can perform the same transformations on polynomial functions that we did with linear and quadratic functions in
Units 1 and 2.
Translations Example: For f(x) = x3 – 6, write the equation given the following transformation.
g(x) = f(x) – 2
h(x) = f(x + 3)
Reflections Example: Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation.
Reflect f(x) across the x-axis.
Reflect f(x) across the y-axis.
Stretch/Compression Example: Let f(x) = 2x4 – 6x2 + 10. Describe g as a transformation of f.
g(x) = ½ f(x)
g(x) = f(3x)
Combination Example 1: Write a function that transforms f(x) = x5 by a vertical shift 2 units up, followed by a
reflection over the x-axis.
Combination Example 2: Write a function that transforms f(x) = 6x3 – 3 by a vertical compression of a factor of 1/3,
then shifts 2 units right.
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