Download 5.3 Notes Beginning Algebra

Chapter 5
Factoring
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Chapter 5-1
Chapter Sections
5.1 – Factoring a Monomial from a Polynomial
5.2 – Factoring by Grouping
5.3 – Factoring Trinomials of the Form
ax2 + bx + c, a = 1
5.4 – Factoring Trinomials of the Form
ax2 + bx + c, a ≠ 1
5.5 – Special Factoring Formulas and a General Review
of Factoring
5.6 – Solving Quadratic Equations Using Factoring
5.7 – Applications of Quadratic Equations
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-2
2
Factoring Trinomials
of the Form
2
ax + bx + c, a ≠ 1
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-3
3
Trial and Error Method
1. Factor out the greatest common factor (GCF), if any.
2. Write all pairs of factors of the coefficient of the
squared term, a.
3. Write all pairs of factors of the constant term, c.
4. Try combinations of these factors until the correct
middle term, bx, is found.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-4
4
Trial and Error Method
Example: Factor 3x2 + 20x + 12.
There is no GCF to factor out.
Since the first term is 3x2, one factor must
contain 3x and the other an x.
(3x + ?)(x + ?)
The product of the last term in the factors
must be 12. Only the positive factors of 12
will be considered.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-5
5
Trial and Error Method
Factor 3x2 + 20x + 12.
Since the product of (3x + 2)and (x + 6) yields the
correct term, 20x, they are the correct factors.
3x2 + 20x + 12 = (3x + 2)(x + 6)
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-6
6
Factor by Grouping Method
1.
2.
3.
4.
Factor out the greatest common factor, if any.
Find two numbers whose product is equal to the
product of a times c, and whose sum is equal to b.
Rewrite the middle term, bx, as the sum or difference
of two terms using the numbers found in step 2.
Factor by grouping as explained in Section 5.2.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-7
7
Factor by Grouping Method
Example: Factor 3x2 + 20x + 12.
There is no factor common to all three
terms.
a=3
b = 20
c = 12
Find two numbers whose product is a · c and
whose sum is b.
Factors of 36 Sum of Factors
(1)(36)
37
(2)(18)
20 
Continued.
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Copyright © 2011 Pearson Education
Chapter 5-8
8
Factor by Grouping Method
Example continued:
Factor 3x2 + 20x + 12.
Use these factors to rewrite 20x.
3x2 + 20x + 12
3x2 + 2x + 18x + 12
Factor by grouping.
3x2 + 2x + 18x + 12 =
x (3x + 2) + 6(3x + 2) =
(3x + 2) (x + 6)
FOIL to check.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-9
9
Factor by Grouping Method
Example: Factor 6x3 + 15x2 – 36x.
Factor out 3x.
6x3 + 15x2 – 36x = 3x (2x2 + 5x – 12)
Rewrite the middle term.
3x (2x2 + 5x – 12) = 3x (2x2 + 8x - 3x – 12)
Factor by grouping.
3x (2x2 + 8x - 3x – 12) = 3x[2x(x +4) - 3(x +4)]
= 3x (x + 4) (2x – 3)
FOIL to check.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Copyright © 2011 Pearson Education
Chapter 5-10
10