Download Solving Polynomial Equations

Solving Polynomial Equations
At the end of this two-part lesson, you will:
 Solve Polynomial Equations by
Factoring/Using Quadratic Formula
 Solve Polynomial Equations by Graphing
X4 – 81 = (x2 + 9) (x + 3) (x-3)
X3 – 125 = 0
(x-5) (x2 +5x + 25)
x = 5; x = ?
RECALL: Quadratic Formula
NOTE: Equation must be in standard
-b+±c =b20 , where
- 4aca, b,
form ax2 + bx
x
=
and c are   and a2a
> 0.
Sums and Differences of Cubes
( u + v) (u2 – uv + v2)
u3 + v3 = ___________________________
u3
-
v3
( u - v) (u2 + uv + v2)
= ___________________________
Review: Factoring Sums and
Differences of Cubes
Being familiar with perfect cubes will make
factoring sums and differences much easier!
x3 – 512
x 3 – 83 =
x3 – 343
x 3 – 73 =
x3 + 125
x 3 + 53 =
x3 + 216
x 3 + 63 =
(x – 8) (x2 + 8x + 64)
(x + 5) (x2 - 5x + 25)
(x – 7) (x2 + 7x + 49)
(x + 6) (x2 - 6x + 36)
SOLVING A POLYNOMIAL
EQUATION
Solve 8x3 + 125 = 0. Find all complex
roots. You will need to use Quadratic
Formula.
( )3 + __3 =
MENTAL MATH: FACTORING HIGHER-DEGREE
Factor
completely.
POLYNOMIALS BY USING
A QUADRATIC
FORM
x4 – 3x2 – 10
x4 + 11x2 + 18
x4 – 8x2 + 12
x4 – 17x2 + 72
x4 – 5x2 + 6
x4 – 16x2 + 63
x4 – 7x2 + 12
x4 + 6x2 + 72
Solving Higher-Degree
Examples Equations by Using a
Quadratic Form
Solve x4 – 6x2 -27 = 0.
Pick your
poison!
Solve x4 – 3x2 - 10 = 0.
Solve x4 + 11x2 +10 = 0.
Solve x4 – 4x2 - 45 = 0.
Example Solving Higher-Degree
Polynomial Equations by Using
Factoring By Grouping
30X3 + 40X2 + 3XPick
+ 4your
=0
poison!
3X4 + 3X2 + 6X2 + 6X = 0
X4 + 12X3 + 4X2 + 48X = 0
…on a final note!
Solving by Graphing
y = x3 + 3x2 – x - 3
You can also solve a
polynomial equation
by graphing each side
of the equation
separately, and
The solutions are -3, 1 & 1.
finding the x-values
3 + 3x2
Step
1:
Graph
y
=
x
1
(zeros) at the point(s)
of intersection.
& y2 = x + 3x
Step 2: Use the intersect
feature to find the x-values
(zeros) of the points of
intersection.
FINAL CHECKS FOR UNDERSTANDING
1. What does it mean for a polynomial with integer
coefficients to be completely factored with
respect to the integers?
2. Give an example of a second-degree
polynomial that cannot be factored with respect
to the integers.
3. Factor completely with respect to the integers:
81x4 – 1
16x3 – 4
1 – 64x3
2x3 – 8x2 + 3x - 12
Homework Assignment:
Pages 324-325. 1-7 odd (calculator)
12-32 all. Column 1 (42-58 all), 61, 62.
Reminder: Chapter 6 Exam on
_____________________.