Algebra II
Name: ________________________
Unit #3: Polynomial Functions
Block: __________
(5.3)
5.3 Solving Polynomial Equations
In words, describe the steps for solving a polynomial equation by factoring: (ex:2𝑥 3 − 5𝑥 2 = 3𝑥)
Solving by Factoring: Finding Real Solutions & Imaginary Solutions
**
9𝑥 2 − 25 = (3𝑥 + 5)(3𝑥 − 5)
***
Think: “SOAP”
Signs for Diff. of Cubes
***
S:
O:
AP:
***NEW: Factoring by Grouping: (**Visual Clue: 4 terms, shared factor)
Example: 2𝑥 3 − 2𝑥 2 − 5𝑥 + 5 = 0
Example:
2𝑥 3 − 8𝑥 2 − 16𝑥 + 64 = 0
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***NEW: Sum or Difference of Cubes: Prove this factoring technique works by expanding the factors
to standard form. Show all steps of your work (distribution)!
(𝒂 + 𝒃)(𝒂𝟐 − 𝒂𝒃 + 𝒃𝟐 )
AND
(𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐 )
Use this factoring technique to solve the
polynomial equations.
S:
Think: “SOAP” Signs for Diff. of Cubes
O:
AP:
**you’ll need the QUADRATIC FORMULA to solve the trinomial factor!
Example: 𝑥 3 − 64 = 0
a = _____ b= _____
Solving Polynomial Equations (with imaginary roots)
a) 𝑥 4 − 3𝑥 2 = 4
b) 𝑥 4 = 16
*Example: 27𝑥 3 + 8 = 0
a = _____ b= _____
c) 𝑥(𝑥 2 + 8) = 8(𝑥 + 1)
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Practice: Solve the polynomial equation by factoring. Then, write the “factoring technique(s)” used.
d) 8x3 + 27 = 0
e) 2x2 – 18 = 0
f) 4x3  500 = 0
g) 3x4  15x2 + 12 = 0
h) x3  12x2 + 11x = 0
i) x3  5x2 + 3x - 15 = 0
j) x4  8x2 + 16 = 0
k) x4  11x2 + 28 = 0
l) x4 + 12x2 = 8x3
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Practice: Solve the polynomial equation by factoring. Then, write the “factoring technique(s)” used.
m) x5 − 3x4 = −2x3
n) x2 + 4 = −4x
o) 2x4 + x3 + 16x + 8 = 0
Writing: When could/should you use factoring techniques to solve a polynomial equation?
Writing: When could/should you use the quadratic formula to solve a polynomial equation?
Writing: How/where will you see your solutions on the graph of the function?
5.2-5.3 Connection: GRAPHING Polynomial Functions!
Graph the polynomial equation using the key features. Then, identify the increasing and decreasing
intervals for the functions (give inequalities about x!) . (you found the zeros on pg. 3-4!)
gg) y = 3x4  15x2 + 12
end behavior: _____________
y-intercept: _____________
zeros: x = -2, -1, 1, 2
multiple zeros: no repeated zeros, all
multiplicity 1
*maximum(s):_____________
*minimum(s): ______________
increasing interval(s):
decreasing interval(s):
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ll) y = x4 – 8x3 + 12x2
end behavior: _____________
y-intercept: _____________
zeros: x = 0, 2, 6
multiple zeros: 0 has a multiplicity of 2, the
zeros 2 and 6 have multiplicity 1
*maximum(s):_____________
*minimum(s): ______________
increasing interval(s):
decreasing interval(s):
mm) y = x5 − 3x4 + 2x3
end behavior: _____________
y-intercept: _____________
zeros: x = 0, 1, 2
multiple zeros: 0 has a multiplicity of 3, and
the zeros 2 and 1 have multiplicity 1
*maximum(s):_____________
*minimum(s): ______________
increasing interval(s):
decreasing interval(s):
Do MathXL 5.3!!
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