Transcript Lesson 5-4: Factoring
Lesson 5-4 & 5-5: Factoring
Objectives: Students will: •Factor using GCF •Identify & factor square trinomials •Identify & factor difference of two squares
• Trinomials
Day 1
Remember: Number of exponent tells you number of Factors/ Solutions/ Roots/ Intercepts x 1 = 1 factor x 2 = 2 factors x 3 = 3 factors x 4 = 4 factors and so on…..
If there is a GCF factor it out!!!!!!
Ex: 2x + 8 2(x + 4)
factor flow chart
Binomial (2 terms) Trinomial (3 terms) Polynomial (4 terms) No Done Is it a difference of squares or cubes ?
A 2 - B 2 or A 3 ± B 3 ex: 4x 2 – 25 or x 3 - 64 Difference of Squares (DS) A 2 - B 2 = (A +B )(A – B) Ex: 4x 2 – 25 = (2x + 5)(2x - 5) yes PST A 2 +2AB+B 2 = (A + B) 2 Or A 2 -2AB+B 2 = (A - B) 2 Ex: 4x 2 –20x +25 = (2x 5) 2 Difference (or sum) of Cubes (A 3 – B 3 ) = (A - B)(A 2 +AB + B 2 ) Or (A 3 + B 3 ) = (A + B)(A 2 - AB + B 2 ) (then factor trinomial if possible) Ex: x 3 – 64 = (x – 4)(x 2 + 4x + 16) Is it a Perfect Square Trinomial? A 2 ± 2AB + B 2 ex: 4x 2 -20x +25 (2x-5) 2 No If a=1 Find ac b If a≠1 Write out factors Rewrite as four terms Repeat with (ax-b) if possible Factor by: Grouping Or Undo foil ( )( ) or box
Factoring The reverse of multiplying 2x(x+3) = 2x 2 + 6x So: 2x 2 + 6x = Look for GCF of all terms → numbers & variables ► Reverse distribute it out → DIVISION Example 1 Factor 6u 2 v 3 – 21uv 2 What is the GCF?
Pull out GCF (divide both terms) 3uv 2 (2uv - 7) 3uv 2
Factoring 4-term Make Sure Polynomial is in descending order!!!!!!!!
3 Methods A) Reverse FOIL F O I L x 2 + 5x + 4x + 20 Find GCF of first two terms- fill first spot Find what makes up ( F) and fill in first spot in other factor already have x so need another x Move to outside (O) already have x so need + 5 Move to inside (I) already have x so need + 4 Check last (L) 4x5 =20 so done!!
REMEMBER: ALWAYS FACTOR A
GCF
1 st IF YOU CAN
• Foil Box x 2 + 5x + 4x + 20 ( x + 5)(x + 4) x + 4 F x 2 I + 4x x + 5 O L + 5x + 20
B) Factor by grouping x 2 + 5x + 4x + 20 x( x + 5 ) + 4(x + 5) (x + 5) (x - 4) Find GCF of first two terms- and factor out Find GCF of second two terms- and factor out What is in parenthesis should match –so factor it out Write what is left as other factor It’s the same either method!!
I like the FOIL method. What do you think????
ax 2 + bx + c – A General Trinomial Where does middle term come from?
(x + 2)(x + 3) = x 2 + 3x + 2x + 6 (2x + 4)(x – 3) = 2x 2 2x 2 - 6x + 4x – 12 - 2x - 12
So to factor we are un
FOIL
ing!!
Steps for General Trinomial Factoring 1) Factor out GCF (always first step) 2) Find product ac that add to b table (to find O and I) 3) Write middle term as combo of factors ( 4 terms) 4)Unfoil or by grouping Example 1: 1) no GCF 2) ac 1*12 1*12 2*6 3*4 b 7 13 8 7 12 x 2 + 7x + 12 x 2 + 4x + 3x + 12 F O I L
TRY Example 2 Factor x 2 – 5x – 24 Example 3 Factor x 2 – 12x + 27
EX 4) Harder One
-24 6x 2 – 5x – 4 6x 2 -8x + 3x - 4 F O I L GCF of first 2
Factor: -7a + 6a
2
-10
Factor: 56 + x – x
2
Assignment (day 1)
• 5-5/227/ 22-72 e
Day 2
Factoring Perfect Squares, Difference of Square, Look back at the forms for each of these from Lesson 5-3
Factor the following
: Ex 1: x 2 – 8x + 16 Perfect Square Trinomial so (x - 4) 2 Ex 2: 9x 2 – 16y 2 Difference of squares so (3x + 4y)(3x – 4y)
Ex 3: Factor 8x 2 – 8y 2 Don’t forget GCF!
Trick: Ex 4: Combo perfect square trinomial and difference of squares x 2 – 2xy + y 2 – 25 Apply PST (x-y) 2 - 25 ( (x-y) + 5 )( (x-y) – 5 ) Now apply DS
Ex 5:
Factor: 64 (
x
2 8
x
16 )
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Marker Board pg 222-223
ASSIGNMENT
• 5-4/222-223/18-62e, 86-92 e
Day 3
• Sum or Difference of cubes
Review Cubing Binomials
• (a+b) 3 = (a+b)(a 2 +2ab+b 2 ) a 3 +3a 2 b+3ab 2 +b 3 (similarly for (a-b) 3 )
Example 1: (a 3 + b 3 ) Notice all the middle terms cancelled out like DS.
What were the terms that cancelled?
What are the factors?
a 2 -ab + b 2 a a 3 -a 2 b ab 2 +b a 2 b -ab 2 b 3 (a 3 + b 3 )= ( a+b)(a 2 -ab+b 2 ) Is the remaining trinomial factorable?
Ex 2: Factor 27x
3
-8y
3 or (3x) 3 _ (2y) 3 9x 2 3x 27x 3 + 6xy 18x 2 y +4y 2 +12xy 2 -2y -18x 2 y -12xy 2
-8y
3 27x 3 -8y 3 =(3x-2y)(9x 2 +6xy+4y 2 ) A 3 – B 3 = (A-B)(A 2 + AB+ B 2 )
Ex 3: Factor x
3
+ 64
Formulas A
3
– B
3
= (A-B)(A
2
+ AB+ B
2
) A
3
+ B
3
= (A+B)(A
2
- AB+ B
2
)
Factor : 125x
3
+1
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