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Unit 4 Lesson 4: Solving Polynomial Equations
State Standards Lesson Goals
3: Add, subtract, multiply and divide (including long division) polynomials.
4: Factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
• Factor polynomial expressions
• Use factoring to solve polynomial equations
ESLRs: Becoming Competent Learners and Complex Thinkers
23 6 27x x
Previously in Math
•Factoring Quadratic Equations▫Use the Distributive Property (look for a
GCF)
▫Check for special case Perfect Square Trinomial Difference of Two Squares
▫“Unfoil” aka “The I-thing”
23 3 9x x 2 9 1x x 22 7 9x x 25x 2 10 25x x 7 7x x 2 49x
Factoring Polynomial Equations
•Sum Of Two Cubes
•Difference of Two Cubes
3 3a b 2 2a b a ab b
3 3a b 2 2a b a ab b
___ ___ ___ ___ ___
___ ___ ___ ___ ___
Are both terms a cube?
Factoring Polynomial Equations
3125 x35
a b 2a 2bab5 x 25 2x5x
Are both terms a cube?
38 343x 32x
a b 2a 2bab2x 7 24x 4914x
3x
37
416 2x x
Factoring Polynomial Equations
32 8 1x x
2 ___ ___ ___ ___ ___x 2x 1 24x 2x 1
3 3 2 2a b a b a ab b
Marker Boards Out!
3 27x 23 3 9x x x
On your marker board
327 64x 23 4 9 12 16x x x
On your marker board
372 243x 39 8 27x
On your marker board
29 2 3 4 12 9x x x
Solving Polynomial Equations
•Factor any way you can.
•Set each factor to zero
•Solve each “mini” equation.
Solving Polynomial Equations3 27 0x
Factor 23 3 9 0x x x
Set each factor to 0 23 0 or 3 9 0x x x
Solve each “mini” equation
23 3 4 1 9
2 13 or x x
3 3 33 or 2ix x X
3 272
3 9 32
Solving Polynomial Equations472 243 0x x
Factor 29 3 4 12 9 0x x x x Set each factor to 0
29 0 or 3 0 or 4 12 9 0x x x x
Solve each “mini” equation
212 4 4 4 9
2 40 or 3 or x x x
3 2 220, 3, i
12 1288
12 64 28
12 8 28i X
Solving Polynomial Equations754 2x x
754 2 0x x
62 27 1 0x x 36 2x x
2 4 22 3 1 9 3 1 0x x x x X22 0 or 3 1 0x x
130 orx x X
3 343 0x
27 7 49 0x x x
On your marker board
Solve:
27 0 or 7 49 0x x x
27 7 4 1 49
2 1x
7 147
2x 7 7 32ix 7x X
354 2 0x
22 3 1 9 3 1 0x x x
On your marker board
Find the real-number solutions of the equation:
27 0 or 7 49 0x x x
27 7 4 1 49
2 1x
7 147
2x 7 7 32ix 7x X
Today’s Assignment
p. 349: 33 – 40, 69, 71, 77, 83, 84
Change form stamp sheet
Check Assignment
p. 348: 33 – 40, 68, 71 77, 83, 84 Change form stamp sheet
22 2 4x x x
24 4 16x x x
26 1 36 6 1x x x
25 2 25 10 4x x x
210 3 100 30 9x x x
227 2 2 4x x x
327 8x
24 2 1 4 2 1x x x
34 8 1x
22 3 3 9x x x
32 27x
2 3 0x x
2 0 or 3 0x x
0 or 3x x
23 3 9 0x x x
23 0 or 3 9 0x x x
3 2723 orx x X
22 1 4 2 1 0x x x
22 1 0 or 4 2 1 0x x x
2 1212 8orx x X
62 16 0x
62 8 0x
2 4 22 2 2 4 0x x x
2 4 22 0 or 2 4x x x
2x iX
Xno real number solutions
7 43 81 0x x
4 33 27 0x x
4 23 3 3 9 0x x x x
4 23 0 or 3 0 or 3 9x x x x 0 or 3 x x
X
Chapter 6 Lesson 4b: Solving Polynomial Equations
State Standards Lesson Goals
3: Add, subtract, multiply and divide (including long division) polynomials.
4: Factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
• Factor polynomial expressions
• Use factoring to solve polynomial equations
ESLRs: Becoming Competent Learners and Complex Thinkers
2 3 2 2y x x
2 23 4y x
Factoring Polynomial Equations
•Factor by Grouping
2 2 2 23 4 12x y x y
2 2 2 23 4 12x y x y 2 2 2 24 3 12x y y x
2 2 23 4 3x y y
Factoring Polynomial Equations
•Recap▫Always look for common factors first
▫Check for Special Patterns Difference of Two Squares Perfect Square Trinomial Sum/Difference of Two Cubes
▫“I-Thing” quadratic equations
▫Look for ways to group
On your marker board
364 27a a
24 3 16 12 9a a a a
464 27a a
On your marker board
2 2 28 16b a ab b
22 4b a b
2 2 3 48 16a b ab b
On your marker board
225x
2 25 6 5 6x x
425 36x
On your marker board
2 22 2bx a b ax
2 2 2 2ax bx a b
2 2x a b a b
2 2a b x
On your marker board
4 24 21x x
2 27 3x x
22x
22 7 9 0x x
Previously in Math
•Solving Quadratic Equations▫Factor
▫Square Root
▫Complete the Square
▫Quadratic Formula
92
2 9 1 0
2 9 0 or 1 0
or 1
x x
x x
x x
2
2
2 32
16
4
x
x
x i
22 7 25x
2
2
2
10 7
10 25 7 25
5 18
5 3 2
5 3 2
x x
x x
x
x
x
2 10 7 0x x
23 3 4 2 5
2 2x
22 3 5 0x x
Solving Polynomial Equations52 18 0y y
Use the distributive property with the common factor 2y
42 9 0y y
Use the Difference of Two Squares Pattern 2 22 3 3 0y y y
Set each factor to 0 2 22 0 or 3 0 or 3 0y y y
Solve each “mini” equation
0 or 3 or 3y y i y
Solving Polynomial Equations3 23 7 12 28x x x
Set equal to zero3 23 7 12 28 0x x x
Group and “undistribute”
2 3 7 4 3 7 0x x x
Gather parentheses 23 7 4 0x x
Factor more? 3 7 2 2 0x x x
Solve 73 , 2, 2
Solving Polynomial Equations3 22 9 18x x x
Set equal to zero3 22 9 18 0x x x
Group and “undistribute”
2 2 9 2 0x x x
Gather parentheses 22 9 0x x
Factor more? 2 3 3 0x x x
Solve 2, 3, 3
Solving Polynomial Equations3 22 3 10 15 0x x x
Group and “undistribute”
2 2 3 5 2 3 0x x x
Gather parentheses 22 3 5 0x x
Factor more?
Solve 32 , 5, 5
Solving Polynomial Equations4 29 12 4 0x x
“unfoil” (this is a perfect square trinomial)
223 2 0x
Square root 23 2 0x
Solve 23 2x
2 23x
63x
Today’s Assignment
p. 349: 42 – 58 e, 70 – 84 eChange from stamp sheet
Check Assignment
p. 349: 42 – 58 e, 70 – 84 Change from stamp sheet
3 210 20 2x x x
210 2 2x x x
22 10 1x x
3 22 4 8x x x
2 2 4 2x x x
22 4x x
3 21 2 4 3 6x x x
21 2 2 3 2x x x
21 2 2 3x x
3 21 2 4 3 6x x x
3 22 2 1x x x
2 2 1 2 1x x x
22 1 1x x
2 24 1 4 1x x
24 1 2 1 2 1x x x
2 29 9x x
2 9 3 3x x x
2 24 9 1x x
22 3 2 3 1x x x
2 29 4 9 4x x
29 4 3 2 3 2x x x
4 23 5 24 0x x
2 23 8 3 0x x
2 28 0 or 3 0x x
2 2 or 3x i x X
3 22 2 0x x x
2 2 1 2 0x x x
22 1 0x x
2 1 1 0x x x
2 0 or 1 0 or 1 0x x x
2, 1, 1
4 22 13 36 0x x
2 22 4 9 0x x
2 2 2 3 3 0x x x x
2, 3
2 3 2 3 0x x x
23 2 0x x
2, 3
3 28 16 0x x x
2 8 16 0x x x
0, 4
24 0x x
4 3 23 3 6 6 0x x x x
3 23 2 2 0x x x x
0, 1, 2
23 1 2 1 0x x x x
23 1 2 0x x x
4 24 20 25 0x x
222 5 0x
102ix
22 5 0x 22 5x 2 5
2x 52ix
X
