5.7 1
Sect. 5.7 Summary of Factoring Techniques General Factoring Strategy:
Z: (if needed) Arrange in Descending Order of Exponents A: (if needed) Factor out the Greatest Common Factor B: Count the number of terms remaining:
2 Terms: If it’s A2-B2 or A3-B3 or A3+B3 use the factoring pattern 3 Terms: If it’s A2+2AB+B2 or A2-2AB+B2 use the factoring pattern
If it’s x2+bx+c then use a factors table If it’s ax2+bx+c then use an ac grouping table
4 Terms: Try grouping terms… 2 and 2, or 3 and 1, or 1 and 3 >4 Terms: Look for binomial or trinomial patterns, and try grouping
C: Always Factor Completely – did you miss a common factor or is there another pattern to be factored?
D: Check your factorization by multiplying
5.7 2
Following the Strategy – 1 Write an equivalent expression by factoring
completely:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
xbxa
baxCheck
babaxbax
xbxa
22
22
22
22
4010
)4(10:
)2)(2(10)4(10
4010
5.7 3
Following the Strategy – 2 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
studentstoleftCheckxxxxxx
xx
x
x
:)42)(2()42()2(
)8)(8(
)2()(
64
22
33
2323
6
5.7 4
Following the Strategy – 3 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
furtherfactoredbetCanyx
yx
')5(7
35726
26
5.7 5
Following the Strategy – 4 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
studentstoleftCheckax
aaxx
aaxx
rearrangeaxax
:)5(2
)2510(2
50202
!20502
2
22
22
22
5.7 6
Following the Strategy – 5 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
!10212...)4)(23(410212...)4)(23(4226...)2)(43(4
583...)1)(83(410)8103(4
3240122
2
yesxxxxxalmostxxxxx
noxxxxxnoxxxxxxforlookingxx
xx
5.7 7
Following the Strategy – 6 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
!)3)(4()3)(4(44,)4()4(3
4123
)12(1)41(3
1243
2
2
2
factorsitaxxaxxxxheyxaxx
axaxx
termsswapaxax
axaxx
5.7 8
Following the Strategy – 7 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
)36)(36(2)3()6(
?93612
?36129
22
22
22
ayaysqsofdiffay
trinsqperfayy
rearrangeyay
5.7 9
Following the Strategy – 8 Factor:
Clues:Common Factors?Binomials?
A2 – B2 A3 + B3 A3 – B3
Trinomials?A2 + 2AB + B2
A2 – 2AB + B2 a=1, use a Factors table a≠1, use an ac Group table
4 or More Terms?Grouping
Factor until PrimeCheck by Multiplying
)()(
))()(())((
)()(
2
22
2222
3223
yxyx
yxyxyxyxyx
yxyyxx
yyxxyx
5.7 10
What Next? Section 5.8
Applications of Polynomial Equations
Look for patterns …
