Alg2 3.5 Notes.notebook
October 17, 2012
3­5 Solving Polynomial Equations
*somewhat combined 3­5 and 3­6 lesson
Skills we need
Factor completely.
1. 2y3 + 4y2 – 30y
2. 3x4 – 6x2 – 24
Solve each equation.
3. x2 – 9 = 0
4. x3 + 3x2 – 4x = 0
Warm­up Answers
Alg2 3.5 Notes.notebook
October 17, 2012
3­5 Solving Polynomial Equations
*somewhat combined 3­5 and 3­6 lesson
Be able to identify multiplicity of roots
Identify all of the real roots of a polynomial
Solve polynomials
To solve a polynomial, and to help graph
I. Dissecting a Polynomial Equation
1.
*example 1 in book
1. Factor
2. Solve
3. Analyze
What are the solutions? (roots)
What general shape does it have?
Where does it cross the x­axis, where does it touch?
Alg2 3.5 Notes.notebook
October 17, 2012
1. continued: 1. Factor
2. Solve
3. Analyze
Roots are x = 0 and x = ­3
What general shape does it have?
Where does it cross the x­axis, where does it touch?
2. 4x6 + 4x5 – 24x4 = 0 1. Factor
2. Solve
3. Analyze
What are the solutions? (roots)
What general shape does it have?
Where does it cross the x­axis, where does it touch?
The roots are 0, –3, and 2. Alg2 3.5 Notes.notebook
October 17, 2012
Looking at a graph doesn't necessarily tell you the multiplicity or the degree.
Take 3x5 + 18x4 + 27x3 = 0. It factors to 3x3(x + 3)(x + 3) = 0
One more time
3. 2x6 – 22x5 + 48x4 + 72x3 = 0
This time, use your graphing calculator to help.
1. Factor
2. Solve
3. Analyze
Alg2 3.5 Notes.notebook
October 17, 2012
II.
4. x3 + 3x2 – 4x – 12 = 0
Factors of –12: ±1, ±2, ±3, ±4, ±6, ±12 Factors of 1: ±1
x = 2, x = ­2, x = 3
Alg2 3.5 Notes.notebook
October 17, 2012
III. Putting it All Together
5.
3­6: Fundamental Theorem of Algebra
1. Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots. The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0
The solutions are 4, –1, 3i, –3i. 2. Write the simplest polynomial with roots –1, , and 4.
Alg2 3.5 Notes.notebook
October 17, 2012
3­5 p.186 #3, 4, 8, 9, 11, 15, 18, 19, 23, 29, 33, 43 ­ 46
*15 problems
Group Quiz Friday:
18 questions
17 are multiple choice
No graphing calculator