aim: factoring course: adv. alg. & trig. aim: renewing an old friendship some more –...
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Aim: Factoring Course: Adv. Alg. & Trig.
Aim: Renewing an old friendship some more – factoring, Algebra! Again!
Do Now:
Factor: 2m2a + 4m3b - 2m4c4x2 – 25
a2 – 8a + 72x2 + 5x + 33x2 - 6x - 24
Aim: Factoring Course: Adv. Alg. & Trig.
Factoring: Greatest Common Factor
Factor by finding the greatest common factor: 2m2a + 4m3b - 2m4c
2m2(a + 2mb - m2c)
GCF - 2m2
General Rules for Factoring with GCF
•Find the greatest common monomial that is a factor of each term by:
•finding the greatest common factor of the coefficients•finding the greatest common factor for each of the variables
•Write the original expression as the product of the greatest common factor and the remaining polynomial
3
y2
3y2GCF
3 y2 (1 – 3y)=Factor: 3y2 – 9y3
Aim: Factoring Course: Adv. Alg. & Trig.
Factoring: Difference between Perfect Squares
-4x2 25
+ -2x 2x5 5
-x2 a2
+ -
1. Are both terms perfect squares?2. Are they subtracted?
x xa a
If yes, then in general terms:x2 - a2 = (x - a)(x + a)
Aim: Factoring Course: Adv. Alg. & Trig.
Factoring Trinomials a = 1
( )( )
F L
a2 - 8a + 7
a a 71 --
O
-7a
+I
-1a
-8a
Aim: Factoring Course: Adv. Alg. & Trig.
Factoring Trinomials a 1
Factor: 2x2 + 5x + 3What do you know?1. The product of 2 binomials
( )( )2. 1st Terms (2x )(x )
3. Last Terms ( 1)( 3)
4. Signs ( + )( + )
Possible Combinations:
(2x + 1)(x + 3) or (2x + 3)(x + 1)
1x
6x7x
3x
2x5x
5x = sum of O & IO
Aim: Factoring Course: Adv. Alg. & Trig.
Factor Completely
3x2 - 6x - 243(x2 - 2x - 8)
3(x - 4)(x + 2)
1. Find the GCF
2. Factor the trinomial
Aim: Factoring Course: Adv. Alg. & Trig.
Factoring - Perfect Cubes
Factoring Difference/Sum of Perfect Cubes
u3 – v3 = (u – v)(u2 + uv + v2)
u3 + v3 = (u + v)(u2 – uv + v2)
Factor:
x3 – 27
x9 + 125
x3 and -27 are perfect cubes:
-27 = (-3)3
u = x and v = (-3)
= (x – (-3))(x2 + x(-3) + (-3)2)
= (x + 3)(x2 – 3x + 9)
x9 and 125 are perfect cubes:
x9 = (x3)3
125 = 53
= (x3 + 5) ((x3)2 – x3(5) + 52)
= (x3 + 5)(x6 – 5x3 + 25)
u = x3 and v = 5
Aim: Factoring Course: Adv. Alg. & Trig.
Model Problems
1. 144 - d2
2. 9x2 - 16y2
3. x4 - y6
4. 8x2 - 8y2
(12 - d)(12 + d)
(9x - 4y)(9x + 4y)
(x2 - y3)(x2 + y3)
8(x - y)(x + y)
8(x2 - y2)
5. s8 - t4 (s4 - t2)(s4 + t2)
(s2 - t)(s2 + t)(s4 + t2)
Note: Check your factoring by multiplying the factors to come up with the original expression.
G.C.F.
Aim: Factoring Course: Adv. Alg. & Trig.
Model Problems
Factor Completely
1. 4a2 + a – 3
2. 8m3 - 1
3. r3 + 3r2 – 54r
4. 4x6 – 4x2
5. 5y5 + 135y2