Transcript Factoring ax2 + bx + c - William H. Peacock, LCDR USN, Ret

Factoring ax2 + bx + c
Section 8-6
Goals
Goal
• To factor trinomials of the
form ax2 + bx + c.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
Factoring
2
ax
+ bx + c
In the previous lesson you factored trinomials of
the form x2 + bx + c. Now you will factor
trinomials of the form ax + bx + c, where a ≠ 0.
X-BOX Method
THE FUN WAY TO FACTOR
TRINOMIALS
X-Box Method
• This is a guaranteed method for factoring
trinomials of the form ax2 + bx +c — no guessing
necessary!
• We will learn how to factor quadratic equations
using the x-box method
• Background knowledge needed:
– Basic x-factor problems
– General form of a trinomial
– The factoring box
The X-Factor
m
n
mn
m n
m+n
Given m & n on the sides of the x factor
m•n is the top of the x factor
m+n is the bottom of the x factor
The X-Factor
mn
m
n
m+n
What’s on Top & Bottom?
5
3
m•n is the top of the x factor
?
5
Given m & n on the sides of the x factor
3
?
m+n is the bottom of the x factor
The X-Factor
5
3
m•n is the top of the x factor
15
5
Given m & n on the sides of the x factor
3
8
m+n is the bottom of the x factor
The X-Factor
-3
9
?
-3
9
?
The X-Factor
-3
9
-27
-3 9
6
The X-Factor
?
12
Given m•n
7
Given m+n
12
7
?
Find m & n
The X-Factor
3
12
Given m•n
7
Given m+n
12
7
4
Find m & n
The X-Factor
?
8
Given m•n
9
Given m+n
8
9
?
Find m & n
The X-Factor
8
8
Given m•n
9
Given m+n
8
9
1
Find m & n
The X-Factor
-84
Given m•n
-5
Given m+n
-84
Find m & n
-5
To solve a difficult X-Factor
START AT 1 AND CHECK THE SUM
Given m•n
-84
Given m+n
-5
-84
-12
Find m & n
7
-5
-84
1 -84 = -83
2 -41 = -39
3 -28 = -25
4 -21 = -17
6 -14 = -8
7 -12 = -5
Why Know the “X”
Factor?
Why Know the “X”
Factor?
The “X” factor is a
Simple way to
FACTOR TRINOMIALS
General Form of a
Trinomial
b is COEFFICIENT
of x
c is COEFFICIENT
of last term
ax  bx  c
2
a is COEFFICIENT
of x2

Identify a, b & c
b is 3
c is 9
a is 5
5x  3x  9
2
Factor the X-Box Method
ax2 + bx + c
Combine the x-factor and a 2⨯2 box to create the x-box method.
Factor Col. 1
Product
First and Last
Coefficients
ac=mn
n
m
Middle
b=m+n
Sum
GCF
Row 1
Factor
Row 2
Factor Col. 2
1st
Term
Factor
n
Factor
m
Last
term
X-Box Procedure
ax2 + bx + c
1.
Create an x-factor with the product ac on the top,
the middle term b on the bottom and the factors m
& n on the sides.
2.
Create a 2x2 box.
– In the top left, put the 1st term. In the bottom right
corner, put the last term.
– Put the two factors m and n times the variable x in the
open boxes.
3.
Determine the GCF of the upper boxes and the
remaining top & side factors.
4.
The sum of the factor’s for the top and the side are
the factors of the polynomial.
1st
Term
Factor
n
Factor
m
Last
term
Factor 5x2+11x+2
5•2=
10
1. Set up & Solve X-factor:
•
•
•
10
1
11
x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
2
5x
5x2
10x
1
x
2
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(x+2)(5x+1)
Factor 3x2 – x – 4
3•-4=
-12
1. Set up & Solve X-factor:
•
•
•
-4
3
-1
3x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-4
x
3x2
-4x
1
3x
-4
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(x+1)(3x – 4)
Factor 12x2 +5x – 2
12•-2=
-24
1. Set up & Solve X-factor:
•
•
•
-3
8
5
4x
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-1
3x 12x2
-3x
2
-2
8x
Factoring
ax2+bx+c
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(3x+2)(4x – 1)
Factor 15x2 +7x – 2
15•-2=
-30
10
1. Set up & Solve X-factor:
•
•
•
-3
7
5x
3x 15x2
2
10x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-1
-3x
-2
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(3x+2)(5x – 1)
Factor 10x2 +9x +2
10•2=
20
5
1. Set up & Solve X-factor:
•
•
•
4
9
5x
2x 10x2
1
5x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
2
4x
2
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(2x+1)(5x +2)
Your Turn:
Factor:
2x² - 13x + 6
Factor 2x2 – 13x +6
2•6=
12
-12
1. Set up & Solve X-factor:
-1
-13
2x
x
2x2
-6 -12x
Factoring
ax2+bx+c
•
•
•
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-1
-x
6
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(2x – 1)(x – 6)
Your Turn:
Factor:
4x² + 11x – 3
Factor 4x²+11x – 3
4•-3=
-12
12
1. Set up & Solve X-factor:
-1
11
4x
Factoring
ax2+bx+c
•
•
•
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-1
x
4x2
-x
3
12x
-3
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(4x – 1)(x + 3)
Your Turn:
Factor:
3x²+8x+4
Factor 3x²+8x+4
4•3=
12
6
1. Set up & Solve X-factor:
•
•
•
2
8
3x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
2
x
3x2
2x
2
6x
4
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(3x + 2)(x + 2)
Your Turn:
Factor:
-5x²+6x-1
Factor -5x²+6x-1
-5•-1=
5
5
1. Set up & Solve X-factor:
•
•
•
1
6
-5x
x
-5x2
-1 5x
Factoring
ax2+bx+c
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
1
x
-1
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(-5x + 1)(x – 1)
Factor Completely
• To factor a polynomial completely, first factor at
the GCF of the polynomial’s terms.
• Then factor the remaining polynomial until it is
written as the product of polynomials that cannot
be factored further.
* Always Factor out the GCF First *
Example: Factor
Completely
Factor Completely: 18x2 – 33x + 12
First factor out the GCF 3
3(6x2 – 11x + 4)
Then factor the remaining polynomial (6x2 – 11x + 4)
Factor 6x2 – 11x+4
6•4=
24
-8
1. Set up & Solve X-factor:
-3
-11
2x
Factoring
ax2+bx+c
•
•
•
Put ac on top
Put b on bottom
Determine side factors
2. Set up & Solve “BOX”:
-1
3x 6x2
-3x
-4 -8x
4
•
•
•
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
3. Determine the GCF of the upper boxes
and the remaining top & side factors.
4. The sum of the factor’s for the top and
the side are the factors of the
polynomial.
(3x – 4)(2x – 1)
Example: Factor
Completely
Factor Completely: 18x2 – 33x + 12
First factor out the GCF 3
3(6x2 – 11x + 4)
Then factor the remaining polynomial (6x2 – 11x + 4)
(3x – 4)(2x – 1)
Completely Factored:
3(3x – 4)(2x – 1)
Your Turn:
1. Completely factor 8x2 – 36x – 20
4(2x + 1)(x – 5)
2. Completely factor 30x2 + 14x – 8
2(3x – 1)(5x + 4)
Assignment
• 8-6 Exercises Pg. 524 - 526: #8 – 58 even