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Factoring Special Products6.46.4
1. Factor perfect square trinomials.2. Factor a difference of squares.3. Factor a difference of cubes.4. Factor a sum of cubes.
Write as many perfect squares as you can.Write as many perfect squares as you can.
Write as many perfect cubes as you can.Write as many perfect cubes as you can.
149
16
25364964
81100121144
169196225625
18
2764
125
Perfect Square Trinomials: Perfect Square Trinomials:
9633 2 xxxx
23x6x is double the product.
22 91243232 yxyxyxyx
232 yx -12xy is double the product.
Perfect squares
Perfect squares
Perfect Square Trinomials: Perfect Square Trinomials:
3615123 2 xxxx
26x15x is not double the product.
Caution: Don’t just check the first and last terms!
Factor completely :Factor completely :
22 25204 baba
252 ba -20ab is double the product.
22 252045252 babababa
Check by foiling!Check by foiling!
Perfect squares
Factor completely :Factor completely :
16920864 2 aa
2138 a-208a is double the product.
Check by foiling!Check by foiling!
Perfect squares
Factor completely :Factor completely :
16249 2 mm
243 m24m is double the product.
1624943 22 mmm
Check by foiling!Check by foiling!
Factor completely :Factor completely :
3662 yy
26y6 is NOT double the product.
PrimePrime
Not a perfect square trinomial.
It may still be factorable.
Factor completely :Factor completely :
247254 2 xx
41296 2 xx
2236 x
Factoring Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Difference of Squares: Difference of Squares:
933 2 xxx
33 xxConjugates
22 943232 yxyxyx
yxyx 3232 Conjugates
Factor completely :Factor completely :
1212 a
1111 aa
Think Conjugates!
Check by foiling!Check by foiling!
Factor:Factor:
1625 2 x
4545 xx
Think Conjugates
Check by foiling!Check by foiling!
Factor completely :Factor completely :
4916 2 x
PrimePrime
The sum of squares CANNOT be factored!The sum of squares CANNOT be factored!
Factor completely :Factor completely :
22 3664 ym
22 9164 ym
Check by foiling!Check by foiling!
ymym 34344
ymym 6868
Factor completely:Factor completely:
164 x
44 22 xx
Check by foiling!Check by foiling!
422 2 xxx
Copyright © 2011 Pearson Education, Inc.
Factoring a Difference of Squares
a2 – b2 = (a + b)(a – b)
Warning: A sum of squares a2 + b2 is prime and cannot be factored.
Sum and Difference of CubesSum and Difference of Cubes
33 yx
Multiply:Multiply:
22 yxyxyx
22 yxyxyx 33 yx
Same.
Cube Root
Opposite.
Square Product Square
Always positive
3 terms – trinomial rather than binomial
Cube Root
a2
Factor completely:Factor completely:
33 278 ba Cubes = trinomial
b3 24a 29b ab6
Square Product Square
y
Factor completely:Factor completely:
643 yCubes = trinomial
4 2y 16 y4
Square Product Square
a10
Factor completely:Factor completely:
33 271000 ba Cubes = trinomial
b3 2100a 29b ab30
Square Product Square
Copyright © 2011 Pearson Education, Inc.
Factoring a Sum or Difference of Cubes
a3 – b3 = (a – b)(a2 + ab + b2)
a3 + b3 = (a + b)(a2 – ab + b2)
Slide 6- 23Copyright © 2011 Pearson Education, Inc.
Factor completely. 9x2 – 49
a) (3x + 5)2
b) (3x + 7)(3x – 7)
c) (3x – 7)2
d) (7x + 3)(7x – 3)
6.4
Slide 6- 24Copyright © 2011 Pearson Education, Inc.
Factor completely. 9x2 – 49
a) (3x + 5)2
b) (3x + 7)(3x – 7)
c) (3x – 7)2
d) (7x + 3)(7x – 3)
6.4
Slide 6- 25Copyright © 2011 Pearson Education, Inc.
Factor completely. 4a2 – 20a + 25
a) (2a + 5)2
b) (2a – 5)2
c) (4a + 5)2
d) (4a – 5)2
6.4
Slide 6- 26Copyright © 2011 Pearson Education, Inc.
Factor completely. 4a2 – 20a + 25
a) (2a + 5)2
b) (2a – 5)2
c) (4a + 5)2
d) (4a – 5)2
6.4
Slide 6- 27Copyright © 2011 Pearson Education, Inc.
Factor completely. 2n2 + 24n + 72
a) 2(n + 6)2
b) 2(n + 6)(n – 6)
c) 2(n – 6)2
d) (2n + 6)(2n – 6)
6.4
Slide 6- 28Copyright © 2011 Pearson Education, Inc.
Factor completely. 2n2 + 24n + 72
a) 2(n + 6)2
b) 2(n + 6)(n – 6)
c) 2(n – 6)2
d) (2n + 6)(2n – 6)
6.4
