Download 5.3 Notes Beginning Algebra

Chapter 5
Factoring
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
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.
Chapter 5-2
2
Factoring Trinomials
of the Form
2
ax + bx + c, a = 1
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-3
3
Factoring Trinomials
Recall that factoring is the reverse process of
multiplication. Using the FOIL method, we can
show that
(x + 3)(x + 4) = x2 + 7x + 12.
Therefore x2 + 7x + 12 = (x + 3)(x + 4)
Note that this trinomial results in the product
of two binomials whose first term is x and
second term is a number (including its sign).
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-4
4
Factoring Trinomials
Factoring any polynomial of the form x2 + bx + c
will result in a pair of binomials:
x2 + bx + c = (x +?)(x +?)
Numbers go here.
L
F
F
( x + 3 )( x + 4 ) =
I
=
O
I
L
x2 + 4x + 3x + 12
x2 +
7x
+ 12
O
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-5
5
Factoring Trinomials
1. Find two numbers whose product equals
the constant, c, and whose sum equals the
coefficient of the x-term, b.
2. Use the two numbers found in step 1,
including their signs, to write the trinomial
in factored form. The trinomial in factored
form will be
(x + first number) (x + second number)
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-6
6
Examples
a.) Factor x2 - 11x - 60.
x2 - 11x - 60 = (x + ?) (x + ?)
Replace the ?s with two numbers that are the
product of -60 and the sum of -11.
x2 + 8x + 15 = (x -15) (x + 4)
b.) Factor x2 + 5x + 12.
This is a prime polynomial because it
cannot be factored.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-7
7
Examples Continued
c.) Factor x2 + 3xy + 2y2.
We must find two numbers whose product is 2
(from 2y2) and whose sum is 3 (from 3xy). The
two numbers and 1 and 2. Thus,
x2 + 3xy + 2y2 = (x + 1y)(x + 2y) =
(x + y)(x + 2y)
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-8
8