5.1 factoring – the greatest common factor finding the greatest common factor: 1.factor – write...
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5.1 Factoring – the Greatest Common Factor
• Finding the Greatest Common Factor:1. Factor – write each number in factored form.2. List common factors3. Choose the smallest exponents – for variables
and prime factors4. Multiply the primes and variables from step 3
• Always factor out the GCF first when factoring an expression
5.1 Factoring – the Greatest Common Factor
• Example: factor 5x2y + 25xy2z
)5(5255
55
525
55
22
0111
12122
01212
yzxxyzxyyx
xyzyxGCF
zyxzxy
zyxyx
5.1 Factoring – Factor By Grouping
• Factoring by grouping1. Group Terms – collect the terms in 2 groups
that have a common factor2. Factor within groups3. Factor the entire polynomial – factor out a
common binomial factor from step 24. If necessary rearrange terms – if step 3 didn’t
work, repeat steps 2 & 3 until you get 2 binomial factors
5.1 Factoring – Factor By Grouping
• Example:
This arrangement doesn’t work.
• Rearrange and try again
)815()65(2
815121022
22
xyyx
xyxyyx
)32)(45(
)23(4)32(5
8121510 22
yxyx
xyyyxx
xyyxyx
5.2 Factoring Trinomials
• Factoring x2 + bx + c (no “ax2” term yet)Find 2 integers: product is c and sum is b
1. Both integers are positive if b and c are positive
2. Both integers are negative if c is positive and b is negative
3. One integer is positive and one is negative if c is negative
5.2 Factoring Trinomials
• Example:
• Example:
)1)(4(
414 ;514
452
xx
xx
)3)(7(
21)3(7 ;437
2142
xx
xx
5.3 Factoring Trinomials – Factor By Grouping
• Factoring ax2 + bx + c by grouping 1. Multiply a times c
2. Find a factorization of the number from step 1 that also adds up to b
3. Split bx into these two factors multiplied by x
4. Factor by grouping (always works)
5.3 Factoring Trinomials – Factor By Grouping
• Example:
• Split up and factor by grouping
62014
)6(20)8(15
)4(30)2(60120
15148 2
b
ac
xx
)52)(34(
)52(3)52(4
15620815148 22
xx
xxx
xxxxx
5.3 More on Factoring Trinomials
• Factoring ax2 + bx + c by using FOIL (in reverse)
1. The first terms must give a product of ax2
(pick two)2. The last terms must have a product of c (pick
two)3. Check to see if the sum of the outer and inner
products equals bx4. Repeat steps 1-3 until step 3 gives a sum = bx
5.3 More on Factoring Trinomials
• Example:
correct 672)2)(32(try
incorrect 682)1)(62(try
incorrect 6132)6)(12(try
?)?)(2(672
2
2
2
2
xxxx
xxxx
xxxx
xxxx
5.3 More on Factoring Trinomials
• Box Method (not in book):
6?
2
?2
?)?)(2(672
2
2
xx
x
xxxx
5.3 More on Factoring Trinomials
• Box Method – keep guessing until cross-product terms add up to the middle value
)2)(32(672 so
642
32
32
2
2
xxxx
x
xxx
x
5.4 Special Factoring Rules
• Difference of 2 squares:
• Example:
• Note: the sum of 2 squares (x2 + y2) cannot be factored.
yxyxyx 22
wwww 3339 222
5.4 Special Factoring Rules
• Perfect square trinomials:
• Examples:
222
222
2
2
yxyxyx
yxyxyx
2222
2222
15152511025
333296
zzzzz
mmmmm
5.4 Special Factoring Rules
• Difference of 2 cubes:
• Example:
2233 yxyxyxyx
)933327 2333 wwwww
5.4 Special Factoring Rules
• Sum of 2 cubes:
• Example:
2233 yxyxyxyx
)933327 2333 wwwww
5.4 Special Factoring Rules
• Summary of Factoring1. Factor out the greatest common factor
2. Count the terms:
– 4 terms: try to factor by grouping
– 3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods
– 2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubes
3. Can any factors be factored further?
5.5 Solving Quadratic Equations by Factoring
• Quadratic Equation:
• Zero-Factor Property:If a and b are real numbers and if ab=0then either a = 0 or b = 0
02 cbxax
5.5 Solving Quadratic Equations by Factoring
• Solving a Quadratic Equation by factoring1. Write in standard form – all terms on one side
of equal sign and zero on the other
2. Factor (completely)
3. Set all factors equal to zero and solve the resulting equations
4. (if time available) check your answers in the original equation
5.5 Solving Quadratic Equations by Factoring
• Example:
1,5.2 :solutions
01or 052
0)1)(52( :factored
0572 :form standard
7522
2
xx
xx
xx
xx
xx
5.6 Applications of Quadratic Equations
• This section covers applications in which quadratic formulas arise.
Example: Pythagorean theorem for right triangles (see next slide)
222 cba
5.6 Applications of Quadratic Equations
• Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
a
b
c
5.6 Applications of Quadratic Equations
• Examplex
x+1
x+2
3
0)1)(3(
032
4412
)2()1(
2
222
222
x
xx
xx
xxxxx
xxx