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a) x 4 – 6x 2 – 27 Example 1 Factor (x 2 + ?)(x 2 – ?) (x 2 + 3)(x 2 – 9) (x 2 + 3)(x – 3)(x + 3) b) x 4 – 3x 2 – 10 (x 2 + ?)(x 2 – ?) (x 2 + 2)(x 2 – 5)
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Warm - upWarm - upFactor:1. 4x2 – 24x 4x(x – 6)
2. 2x2 + 11x – 21 (2x – 3)(x + 7)
3. 4x2 – 36x + 81 (2x – 9)2
Solve:4. x2 + 10x + 25 = 0
x = -55. 6x2 + x = 15 x = 3/2 and -5/3
2x2x22 – 5x – 12 – 5x – 12 We know how to factor:We know how to factor:- A General Trinomial- A General Trinomial
Solving Polynomial Solving Polynomial EquationsEquations
(2x + 3)(x – 4) (2x + 3)(x – 4) - A Perfect Square Trinomial- A Perfect Square Trinomial xx22 + 10x + 25 + 10x + 25
(x + 5)(x + 5) = (x +5)(x + 5)(x + 5) = (x +5)22
- The Difference of two Squares- The Difference of two Squares xx22 – 9 – 9 (x)(x)2 2 – 3– 322
(x + 3)(x – 3)(x + 3)(x – 3)- A Common Monomial Factor- A Common Monomial Factor 6x6x2 2 + 15x + 15x
3x(2x + 5)3x(2x + 5)
a) x4 – 6x2 – 27
Example 1Example 1FactorFactor
(x2 + ?)(x2 – ?)(x2 + 3)(x2 – 9)(x2 + 3)(x – 3)(x + 3)
b) x4 – 3x2 – 10(x2 + ?)(x2 – ?)(x2 + 2)(x2 – 5)
aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))
aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))
Sum of Two CubesSum of Two Cubes ** Special Factoring Patterns** Special Factoring Patterns
ex. xex. x33 + 8 + 8 a = xa = x
(x + 2)(x(x + 2)(x22 – 2x + 4) – 2x + 4)
aa33 – b – b33 = (a – b)(a = (a – b)(a22 + ab + b + ab + b22))
Example 2Example 2xx3 3 + 125 + 125
xx33 + 5 + 53 3
Difference of Two CubesDifference of Two Cubes
b = 2b = 2
ex. 8xex. 8x33 – 1 – 1
xx33 + 2 + 233
a = 2xa = 2x
(2x – 1)(4x(2x – 1)(4x22 + 2x + 1) + 2x + 1) b = 1b = 1
(2x)(2x)33 – (1) – (1)33
= (x + 5)(x= (x + 5)(x22 – 5x + 25) – 5x + 25)
a) x3 – 27
Example 3Example 3FactorFactor
aa33 – b – b33 = (a – b)(a = (a – b)(a22 + ab + b + ab + b22))
xx33 – 3 – 33 3 = (x – 3)(x= (x – 3)(x22 + 3x + 9) + 3x + 9)
b) 8x3 + 64aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))
(2x)(2x)33 + (4) + (4)3 3 = (2x + 4)(4x= (2x + 4)(4x22 – 8x + 16) – 8x + 16)
Must be the sameMust be the same
x2(x – 2)
x3 – 2x2 – 9x + 18
(x2 – 9)(x – 2)
Extra Example 2Extra Example 2Factor by groupingFactor by grouping
-9(x – 2)
(x – 3)(x + 3)(x – 2)
Solving Polynomial EquationsSolving Polynomial Equations1. Factor out GCF2. Factor remaining
quadratic equation1. If remaining
equation can not be factored, use quadratic formula.
3. Solve all equations for variable.
01223 xxx#1: 0122 xxx
034 xxx0x 04x
4x03x
3x
0189 234 xxx#3:
018922 xxx 0362 xxx02 x 06x
6x03x3x0x
xxx 44 23 #7:24x24x
xxx 44 23 x4x4
044 23 xxx 0442 xxx 022 xxx0x 02x
2x02x
2x
045 24 xx#6:
014 22 xx
22 xx 011 xx02x
2x02x
2x01x1x
01x1x
03 23 xxx#1B:
0132 xxx0x 0132 xx
What do we do when we can’t factor?
131
cba
aacbbx
242
12
11433 2 x
253
x
Roots:
253,
253,0
010142 345 xxx#4B:
0572 23 xxx02 3 x 0572 xx
What do we do when we can’t factor?
571
cba
aacbbx
242
12
51477 2 x
2297
x
Roots:
2297,
2297,0
0x
08208 23 xxx#1M:
02524 2 xxx04 x 0252 2 xx
What do we do when we can’t factor?
252
cba
aacbbx
242
22
22455 2 x
435
495
x
Roots:
0x
248
435
x
21
42
435
x
21,2,0
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