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Warm Up
1. Multiply using the area model or distribution:
3x2( x + 5)
2. Use the Area Model to multiply the following:
(2x + 4) (x - 2)
Math I
UNIT QUESTION: In what ways can
algebraic methods be used in
problems solving?
Standard: MM1A2
Today’s Question:
How do we find the GCF of a
polynomial?
Standard: MM1A2f
3 EASY PIECES
Factor “T” & Factor “X” &
Factor Box
This is a guide for
FACTORING POLYNOMIALS
What is Factoring?
What is Factoring?
To understand factoring we’ll first break down what
multiplying is:
MULTIPLYING:
(MONOMIAL)• ( MONOMIAL) = PRODUCT
OR
(FACTOR)• ( FACTOR) = PRODUCT
What is the first thing you should
check when factoring polynomials?
What is the first thing you should
check when factoring polynomials?
Are there any common factors
(numbers or variables) with
All of the terms?
In other words does a
GCF exist?
What is the GCF?
25x 5x
2
Use a Factor “T”
25x 5x
2
A factor “T” is a method of
Finding the GCF
Use a Factor “T”
5x 25x 5x
2
Find the greatest
common # & variable
Use a Factor “T”
5x 1
5x 25x 5x
2
Divide the GCF into each
term to complete factoring
Do A Mental Check
5x 1
5x 25x 5x
2
Mental Check
5x(5x 1) 25x 5x
2
Mental Check
Mental Check
Use this to indicate you have
Completed a mental check.
We can represent this problem as
a Factor Tree
25x 5x
2
5x • 5x 1
Factor 10x 20x
3 2
GCF Factor “T”
Factor 10x 20x
3 2
GCF Factor “T”
1. Find Greatest Common Number
2. Find Greatest Common Variable
10x 10x 20x
2 3 2
Factor 10x 20x
3 2
GCF Factor “T”
(x 2) 1. Find Greatest Common Number
2. Find Greatest Common Variable
10x 10x 20x
2 3 2
3. Divide GCF into each Term
Factor 10x 20x
3 2
GCF Factor “T”
(x 2) 1. Find Greatest Common Number
2. Find Greatest Common Variable
10x 10x 20x
2 3 2
3. Divide GCF into each Term
Mental Check 4. Do a Mental Check
10x2(x-2)
Factor
3x 9x 15x
3 2
GCF Factor “T”
Factor
3x 9x 15x
3 2
GCF Factor “T”
1. Find Greatest Common Number
2. Find Greatest Common Variable
3x 3x 9x 15x
3 2
Factor
3x 9x 15x
3 2
GCF Factor “T”
(x 3x 5)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
3x 3x 9x 15x
3 2
3. Divide GCF into each Term
Factor
3x 9x 15x
3 2
GCF Factor “T”
(x 3x 5)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
3x 3x 9x 15x
3 2
3. Divide GCF into each Term
Mental Check 4. Do a Mental Check
3x(x 3x 5)
2
Factor
2 2
6ab15a b GCF Factor “T”
Factor
2 2
6ab15a b GCF Factor “T”
1. Find Greatest Common Number
2. Find Greatest Common Variable
3ab 6ab15a b
2 2
Factor
2 2
6ab15a b GCF Factor “T”
1. Find Greatest Common Number
(2 5ab)
2. Find Greatest Common Variable
3ab 6ab15a b2 2
3. Divide GCF into each Term
Factor
2 2
6ab15a b GCF Factor “T”
1. Find Greatest Common Number
(2 5ab)
2. Find Greatest Common Variable
3ab 6ab15a b2 2
3. Divide GCF into each Term
Mental Check 4. Do a Mental Check
3ab(2 5ab)
Factor
7x 28x 35x
3 2
GCF Factor “T”
Factor
7x 28x 35x
3 2
GCF Factor “T”
1. Find Greatest Common Number
2. Find Greatest Common Variable
7x 7x 28x 35x
3 2
Factor
7x 28x 35x
3 2
GCF Factor “T”
(x 4x 5)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
7x 7x 28x 35x
3 2
3. Divide GCF into each Term
Factor
7x 28x 35x
3 2
GCF Factor “T”
(x 4x 5)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
7x 7x 28x 35x
3 2
3. Divide GCF into each Term
Mental Check 4. Do a Mental Check
7x (x 4x 5)
2
Factor
4s 8s 36
2
GCF Factor “T”
Factor
4s 8s 36
2
GCF Factor “T”
1. Find Greatest Common Number
2. Find Greatest Common Variable
4 4s 8s 36
2
Factor
4s 8s 36
2
GCF Factor “T”
(s 2s 9)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
4 4s 8s 36
2
3. Divide GCF into each Term
Factor
4s 8s 36
2
GCF Factor “T”
(s 2s 9)
2 1. Find Greatest Common Number
2. Find Greatest Common Variable
4 4s 8s 36
2
3. Divide GCF into each Term
Mental Check 4. Do a Mental Check
4(s 2s 9)
2
GCF Factoring Practice
1. 25x 15
2. - 36x 12x
3. 44y 5 11y 2 88y
4. 5x 7 y 4 x 9 y 7
5. 6x 2 y 3 z 3xyz
GCF Factoring Practice
1. 25x 15 5(5x + 3)
2. - 36x 2 12x 12x(-x+1)
3. 44y 5 11y 2 88y 11y(4y4 + y - 8)
4. 5x 7 y 4 x 9 y 7 x y (5 x y )
7 4 2 3
5. 6x y z 3xyz
2 3
3xyz(2xy 2 1)
Factoring 4 Term Polynomials
with Factor Box
Factoring 4 Term
x3+3x2+2x+6 Polynomials
Factoring 4 Term
x3+3x2+2x+6 Polynomials
1. Put 4 terms in boxes in
x3 3x2 1 2 descending
order.
3 4
2x 6
Factoring 4 Term
x3+3x2+2x+6 Polynomials
1. Put 4 terms in boxes in
x2 x3 3x2 1 2 descending
order.
3 4
2x 6 2. Use a Factor “T” to find
GCF of upper boxes.
Factoring 4 Term
x3+3x2+2x+6 Polynomials
x +3 1. Put 4 terms in boxes in
x2 x3 3x2 1 2 descending
order.
3 4
2x 6 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
Factoring 4 Term
x3+3x2+2x+6 Polynomials
x +3 1. Put 4 terms in boxes in
x2 x3 3x2 1 2 descending
order.
3 4
+2 2x 6 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
Factoring 4 Term
x3+3x2+2x+6 Polynomials
x +3 1. Put 4 terms in boxes in
x3 3x2 1 2 descending
x2 order.
3 4
+2 2x 6 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
=(x+3)(x2+2) 5. Answer is Top Factor
times Side Factor
Factoring 4 Term
x3+x2+4x+4 Polynomials
Factoring 4 Term
x3+x2+4x+4 Polynomials
1. Put 4 terms in boxes in
x3 x2 1 2 descending
order.
3 4
4x 4
Factoring 4 Term
x3+x2+4x+4 Polynomials
1. Put 4 terms in boxes in
x3 x2 1 2 descending
x2 order.
3 4
4x 4 2. Use a Factor “T” to find
GCF of upper boxes.
Factoring 4 Term
x3+x2+4x+4 Polynomials
x +1 1. Put 4 terms in boxes in
x3 x2 1 2 descending
x2 order.
3 4
4x 4 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
Factoring 4 Term
x3+x2+4x+4 Polynomials
x +1 1. Put 4 terms in boxes in
x3 x2 1 2 descending
x2 order.
3 4
+4 4x 4 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
Factoring 4 Term
x3+x2+4x+4 Polynomials
x +1 1. Put 4 terms in boxes in
x3 x2 1 2 descending
x2 order.
3 4
+4 4x 4 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
=(x+1)(x2+4) 5. Answer is Top Factor
times Side Factor
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
1. Put 4 terms in boxes in
8x3 10x2 1 2 descending
order.
3 4
-12x -15
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
1. Put 4 terms in boxes in
8x3 10x2 1 2 descending
2x2 order.
3 4
-12x -15 2. Use a Factor “T” to find
GCF of upper boxes.
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
4x +5 1. Put 4 terms in boxes in
8x3 10x2 1 2 descending
2x2 order.
3 4
-12x -15 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
4x +5 1. Put 4 terms in boxes in
8x3 10x2 1 2 descending
2x2 order.
3 4
-3 -12x -15 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
Factoring 4 Term
8x3+10x2-12x-15 Polynomials
4x +5 1. Put 4 terms in boxes in
8x3 10x2 1 2 descending
2x2 order.
3 4
-3 -12x -15 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
=(4x+5)(2x2-3) 5. Answer is Top Factor
times Side Factor
Factoring 4 Term
15x3-25x2-3x+5 Polynomials
1. Put 4 terms in boxes in
1 2 descending
order.
3 4
2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
5. Answer is Top Factor
times Side Factor
Factoring 4 Term
15x3-25x2-3x+5 Polynomials
3x -5 1. Put 4 terms in boxes in
15x3 -25x2 1 2 descending
order.
5x2 3 4
-1 -3x 5 2. Use a Factor “T” to find
GCF of upper boxes.
3. Complete Top Factor
4. Complete Side Factor.
=(3x-5)(5x2-1) 5. Answer is Top Factor
times Side Factor
Practice 4 Term Factoring
1. 2x3-16x2-11x+88
2. 6y5+18y3-2y2-6
3. 4x6+36x4+4x2+36
4. 14x3+10x2-7x-5
Practice 4 Term Factoring
1. 2x3-16x2-11x+88 =(x-8)(2x2-11)
2. 6y5+18y3-2y2-6 =(6y3-2)(y2+3)
3. 4x6+36x4+4x2+36 =(4x4+4)(x2+9)
4. 14x3+10x2-7x-5 =(7x+5)(2x2-1)
GCF & 4 TERM POLYNOMIALS PRACTICE SHEET
End of Day 1….
The Factor “X”
a b Given a & b on the sides of the x factor
ab a•b is the top of the x factor
a b
a+b a+b is the bottom of the x factor
FACTOR “X”
ab
a b
a+b
What’s on Top & Bottom?
5 3 Given a & b on the sides of the x factor
? a•b is the top of the x factor
5 3
? a+b is the bottom of the x factor
The Factor “X”
5 3 Given a & b on the sides of the x factor
15 a•b is the top of the x factor
5 3
8 a+b is the bottom of the x factor
The Factor “X”
-3 9
?
-3 9
?
The Factor “X”
-3 9
-27
-3 9
6
The Factor “X”
-5 -6
?
-5 -6
?
The Factor “X”
-5 -6
30
-5 -6
-11
The Factor “X”
12 Given a•b
7 Given a+b
12
? ? Find a & b
7
The Factor “X”
12 Given a•b
7 Given a+b
12
3 4 Find a & b
7
The Factor “X”
8 Given a•b
Given a+b
9
8
? ? Find a & b
9
The Factor “X”
8 Given a•b
Given a+b
9
8
8 1 Find a & b
9
The Factor “X”
-84 Given a•b
Given a+b
-5
-84
Find a & b
-5
To solve a difficult “X” factor
START AT 1 AND CHECK THE SUM
-84
-84 Given a•b
Given a+b
1 -84 = -83
-5 2 -41 = -39
3 -28 = -25
-84 Find a & b 4 -21 = -17
-12 7 6 -14 = -8
-5 7 -12 = -5
Complete These Factor X’s
21 36
10 12
18 36
9 -12
Complete These Factor X’s
-15 56
2 15
-9 55
-8 -16
Complete These Factor X’s
-60 -20
4 1
18 -14
-19 -5
Why Know the “X” Factor?
Why Know the “X” Factor?
The “X” factor is a
Simple way to
FACTOR TRINOMIALS
Factoring Trinomials
c is COEFFICIENT
b is COEFFICIENT
of last term
of x
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
Identify a, b & c
c is -9
a is 25
25x 9
2
b is 0
Identify a, b & c
1. 4x2+7x+1 a= b= c=
2. 2x2+21x+3 a= b= c=
3. x2-1 a= b= c=
4. -x2+8x-6 a= b= c=
5. 5x2+x+2 a= b= c=
6. x2+9yx+y2 a= b= c=
7. 49x2-16 a= b= c=
Identify a, b & c
1. 4x2+7x+1 a=4 b=7 c=1
2. 2x2+21x+3 a=2 b=21 c=3
3. x2-1 a=1 b=0 c=-1
4. -x2+8x-6 a=-1 b=8 c=-6
5. 5x2+x+2 a=5 b=1 c=2
6. x2+9yx+y2 a=1 b=9 c=1
7. 49x2-16 a=49 b=0 c=-16
Identify a, b & c
Name any Perfect Squares
8. 4x2+16xy+16y2 a= b= c=
9. 25x2+30x+9 a= b= c=
10. 4a2 b2 -60ab+225 a= b= c=
11. x2-36 a= b= c=
12. 9x2-66x+121 a= b= c=
13. x2+2yx+y2 a= b= c=
14. 100x2-81 a= b= c=
Identify a, b & c
Name any Perfect Squares
8. 4x2+16xy+16y2 a=4 b=16 c= 16
9. 25x2+30x+9 a=25 b=30 c=9
10. 4a2 b2 -60ab+225 a= 4 b= -60 c=225
11. x2-36 a=1 b=0 c= -36
12. 9x2-66x+121 a=9 b= -66 c=121
13. x2+2yx+y2 a=1 b=2 c= 1
14. 100x2-81 a=100 b=0 c=81
Set up & Solve Factor Factor “X” for
“X” for x2+5x+6 Trinomials
a=1, b=5, c=6 1. IDENTIFY a, b & c
2. Put ac on top
a•c 3. Put b on bottom
6 4. Determine side factors
(Start at 1 and check the
2 3 sum)
5
b
Name ____________________________________
Set up & Solve "X" Factor Per. ___________
1. Set up & Solve Factor "X" for : 2. Set up & Solve Factor "X" for : 3. Set up & Solve Factor "X" for :
Set up & Solve Factor Factor “X” for
“X” for x2+5x+6 Trinomials
2 2 2
a=1, b=5, c=6 1.
2.
IDENTIFY a, b & c
Put ac on top
5x +9x+9 3x -16x-2 3x +x-4
a•c 3. Put b on bottom a= b= c= a= b= c= a= b= c=
6 4. Determine side factors
(Start at 1 and check the
2 3 sum)
5
b
4. Set up & Solve Factor "X" for : 5. Set up & Solve Factor "X" for : 6. Set up & Solve Factor "X" for : 7. Set up & Solve Factor "X" for :
x2+10x+25 3x2+6x-9 2x2-6X-8 2x2+20x+18
a= b= c= a= b= c= a= b= c= a= b= c=
8. Set up & Solve Factor "X" for : 9. Set up & Solve Factor "X" for : 10. Set up & Solve Factor "X" 11. Set up & Solve Factor "X"
2 2
2
2x +x-10
2
4x -6x-10 for: x -64 for: 6x -1
a= b= c= a= b= c= a= b= c= a= b= c=
End of Day 2:
X-BOX
THE FUN WAY TO FACTOR
TRINOMIALS
Factoring ax2+bx+c
Factor x2+5x+6 Polynomials with X-BOX
Factoring ax2+bx+c
Factor x2+5x+6 Polynomials
1•6= 1. Set up & Solve “X” factor:
6 •Put ac on top
2 3 •Put b on bottom
•Determine side factors
5
Factoring ax2+bx+c
Factor x2+5x+6 Polynomials
6 1. Set up & Solve “X” factor:
•Put ac on top
3 2 •Put b on bottom
•Determine side factors
5
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
x2 2x •Put Side Factors Times x in Box
3x 6
Factoring ax2+bx+c
Factor x2+5x+6 Polynomials
6 1. Set up & Solve “X” factor:
•Put ac on top
2 3 •Put b on bottom
•Determine side factors
5
2. Set up & Solve “BOX”:
x 2 •Put First Term in First Box
•Put Last Term in Last Box
x x2 2x •Put Side Factors Times x in Box
•Determine GCF of upper boxes
and top&side factors for
3 3x 6 ANSWER
=(x+2)(x+3)
Check Out the Answer
x2+5x+6 = (x+2)(x+3)
x 2
x2+5x+6
x x2 2x
(x+2) (x+3) 3 3x 6
FACTOR TREE MODEL
Multiplying (x+2)(x+3)
should get x2+5x+6
Final Answer is
Top & Side Factors
x 2
x x2 2x
3 3x 6
Get top & side factors with
a factor “T”
Factoring ax2+bx+c
Factor x2+2x-8 Polynomials
Factoring ax2+bx+c
Factor x2+2x-8 Polynomials
-8 1. Set up & Solve “X” factor:
•Put ac on top
-2 4 •Put b on bottom
•Determine side factors
2
Factoring ax2+bx+c
Factor x2+2x-8 Polynomials
-8 1. Set up & Solve “X” factor:
•Put ac on top
-2 4 •Put b on bottom
•Determine side factors
2
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
x2 -2x •Put Side Factors Times x in Box
4x -8
Factoring ax2+bx+c
Factor x2+2x-8 Polynomials
-8 1. Set up & Solve “X” factor:
•Put ac on top
-2 4 •Put b on bottom
•Determine side factors
2
2. Set up & Solve “BOX”:
x -2 •Put First Term in First Box
•Put Last Term in Last Box
x x2 -2x •Put Side Factors Times x in Box
•Determine GCF of upper boxes
and top & side factors for ANSWER
4 4x -8
=(x-2)(x+4)
Factoring ax2+bx+c
Factor 3y2-y-4 Polynomials
Factoring ax2+bx+c
Factor 3y2-y-4 Polynomials
-12 1. Set up & Solve “X” factor:
•Put ac on top
3 -4 •Put b on bottom
•Determine side factors
-1
Factoring ax2+bx+c
Factor 3y2-y-4 Polynomials
-12 1. Set up & Solve “X” factor:
•Put ac on top
3 -4 •Put b on bottom
•Determine side factors
-1
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
3y2 -4y •Put Side Factors Times x in Box
3y -4
Factoring ax2+bx+c
Factor 3y2-y-4 Polynomials
-12 1. Set up & Solve “X” factor:
•Put ac on top
3 -4 •Put b on bottom
•Determine side factors
-1
2. Set up & Solve “BOX”:
3y -4 •Put First Term in First Box
•Put Last Term in Last Box
y 3y2 -4y •Put Side Factors Times x in Box
•Determine GCF of upper boxes
and top & side factors for ANSWER
1 3y -4
=(y+1)(3y-4)
Factoring ax2+bx+c
Factor 12x2+5x-2 Polynomials
Factoring ax2+bx+c
Factor 12x2+5x-2 Polynomials
-24 1. Set up & Solve “X” factor:
•Put ac on top
-3 8 •Put b on bottom
•Determine side factors
5
Factoring ax2+bx+c
Factor 12x2+5x-2 Polynomials
-24 1. Set up & Solve “X” factor:
•Put ac on top
-3 8 •Put b on bottom
•Determine side factors
5
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
12x2 8x •Put Side Factors Times x in Box
-3x -2
Factoring ax2+bx+c
Factor 12x2+5x-2 Polynomials
-24 1. Set up & Solve “X” factor:
•Put ac on top
-3 8 •Put b on bottom
•Determine side factors
5
2. Set up & Solve “BOX”:
3x +2 •Put First Term in First Box
•Put Last Term in Last Box
4x 12x2 8x •Put Side Factors Times x in Box
•Determine GCF of upper boxes
and top & side factors for ANSWER
-1 -3x -2
=(3x+2)(4x-1)
Factoring ax2+bx+c
Factor x2-25 Polynomials
Factoring ax2+bx+c
Factor x2-25 Polynomials
-25 1. Set up & Solve “X” factor:
•Put ac on top
-5 5 •Put b on bottom
•Determine side factors
0
Factoring ax2+bx+c
Factor x2-25 Polynomials
-25 1. Set up & Solve “X” factor:
•Put ac on top
-5 5 •Put b on bottom
•Determine side factors
0
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
x2 5x •Put Side Factors Times x in Box
-5x -25
Factoring ax2+bx+c
Factor x2-25 Polynomials
-25 1. Set up & Solve “X” factor:
•Put ac on top
-5 5 •Put b on bottom
•Determine side factors
0
2. Set up & Solve “BOX”:
x +5 •Put First Term in First Box
•Put Last Term in Last Box
x x2 5x •Put Side Factors Times x in Box
3. Determine GCF of upper boxes
and top & side factors for ANSWER
-5 -5x -25
=(x+5)(x-5)
FACTOR POLYNOMIALS PRACTICE SHEET If a factor “X” doesn’t have a solution NAME ______________
the polynomial is unfactorable PER______________
Factoring ax 2 + bx +c 2
Factor 3y 2 -y-4 Pol yn om i al s
Qu ic kTime ™ and a
TIFF (Unco mpres sed ) d eco mpres sor
are n eed ed to se e th i s pi cture. 1. x +5x-50
-12 1. Set up & Solve “X ” factor:
3 -4
•Put ac on top
•Put b on bottom
a= b= c=
•Determine side factors
-1 2. Set up & Solve “BOX ”:
•Put First Term in First Box
3y -4 •Put Last Term in Last Box
•Put Side Factors Times x in Box
y 3y 2 -4y •Determine GCF of upper boxes
and top & side factors for ANSWER
1 3y -4 (________)(_________)
=(y+1)(3y-4)
2 2
2. 8x +18x+4 3. 2x -x-10
a= b= c= a= b= c=
(________)(_________) (________)(_________)
2 2
4. x -7x-18 5. 4x +3x-7
a= b= c= a= b= c=
(________)(_________) (________)(_________)
Factoring ax2+bx+c
Factor 4y2-49 Polynomials
Factoring ax2+bx+c
Factor 4y2-49 Polynomials
-196 1. Set up & Solve “X” factor:
•Put ac on top
•Put b on bottom
•Determine side factors
0
Factoring ax2+bx+c
Factor 4y2-49 Polynomials
-196 1. Set up & Solve “X” factor:
•Put ac on top
-14 14 •Put b on bottom
•Determine side factors
0
Factoring ax2+bx+c
Factor 4y2-49 Polynomials
-196 1. Set up & Solve “X” factor:
•Put ac on top
-14 14 •Put b on bottom
•Determine side factors
0
2. Set up & Solve “BOX”:
•Put First Term in First Box
•Put Last Term in Last Box
4y2 14y •Put Side Factors Times x in Box
-14y -49
Factoring ax2+bx+c
Factor 4y2-49 Polynomials
-196 1. Set up & Solve “X” factor:
•Put ac on top
-14 14 •Put b on bottom
•Determine side factors
0
2. Set up & Solve “BOX”:
2y +7 •Put First Term in First Box
•Put Last Term in Last Box
2y 4y2 14y •Put Side Factors Times x in Box
•Determine GCF of upper boxes
and top & side factors for ANSWER
-7 -14y -49
=(2y+7)(2y-7)
Recognizing Monomial Squares
Numbers that are perfect squares are: 1, 4, 9, 16, 25, 36,
49, 64, 81, 100, 121, 144, 169, 196, 225,…(Know These)
Variables that are perfect squares are: x2, a4, y22, x100…
(Any even powered variable is a perfect square)
3x25 EVEN POWERED
EXPONENTS ARE
3x25 9x50 SQUARES
Factoring Binomials
All Binomials can be factored with the X-Box
method where b=0, so the “X” factor looks like:
-196
-14 14
0
Another method is called Difference of Squares.
This method is easy to use. The X-Box is sometimes
difficult when a•c is large.
DIFFERENCE OF SQUARES
Method of Factoring Binomials
A binomial can only be factored (other than GCF) if
it is a DOS - Difference of Squares
Both Monomials must be Monomial Squares and the
Difference in Math is a Negative.
DIFFERENCE
Monomial Square
Monomial Square
x2 - 25
DOS Formula
x2 - 25 = (x+5)(x-5) or
r2 - s2 = (r+s)(r-s)
r is the square root s is the square root
of the 1st term of the 2nd term
Factoring a
Factor x2-16 Difference of Squares
Factoring a
Factor x2-16 Difference of Squares
x 16
2 1. Recognize Both Terms are Squares
(r is the sq. root of the first square)
r=x s=4 (s is the sq. root of the second square)
(One term must be negative for Diff.)
Factoring a
Factor x2-16 Difference of Squares
x 16
2 1. Recognize Both Terms are Squares
(r is the sq. root of the first square)
r=x s=4 (s is the sq. root of the second square)
(One term must be negative for Diff.)
2. Use the Diff. Of Squares Formula
(x 4)(x 4) r s (r s)(r s)
2 2
Factoring a
Factor 16x4-36 Difference of Squares
Factoring a
Factor 16x4-36 Difference of Squares
16x 36
4 1. Recognize Both Terms are Squares
(r is the sq. root of the first square)
r=4x2 s=6 (s is the sq. root of the second square)
Factoring a
Factor 16x4-36 Difference of Squares
DIFFERENCE
16x 36
4 1. Recognize Both Terms are Squares
(r is the sq. root of the first square)
r=4x2 s=6 (s is the sq. root of the second square)
2. One term must be negative for Diff.
Factoring a
Factor 16x4-36 Difference of Squares
16x 36
4 1. Recognize Both Terms are Squares
(r is the sq. root of the first square)
r=4x2 s=6 (s is the sq. root of the second square)
2. One term must be negative for Diff.
(4x 2 6)(4x 2 6) 3. Use the Diff. Of Squares Formula
r 2 s2 (r s)(r s)
If a binomial is not a DOS, it is
non-factorable (except GCF)
What is a Trinomial Square?
A Trinomial Square is a Perfect
Square that is a Trinomial and
The Square Root is a Binomial.
x+4 x2+8x+16
x+4
(x+4)2= x2+8x+16
How to Recognize a
TRINOMIAL SQUARE
Easy and convenient way to factor a Trinomial Square.
The First & Last terms are squares and the middle term is
plus or minus 2 • sq. root of 1st term • sq. root of 2nd term
x2 ± 10x + 25
Monomial Square Monomial Square
±2•r•s
r2 s2
Formula for Factoring a
TRINOMIAL SQUARE
r2±2rs+s2 = (r ± s)2
x2 + 10x + 25 = (x+5)2
Monomial Square
r2 Monomial Square
s2
±2•r•s
Plus or Minus
TRINOMIAL SQUARE
r2±2rs+s2 = (r ± s)2
x2 + 10x + 25 = (x+5)2
x2 - 10x + 25 = (x-5)2
IF MIDDLE TERM IS PLUS ANSWER IS PLUS
IF MIDDLE TERM IS MINUS ANSWER IS MINUS
Factoring a
Factor 4x2+24x+36 Trinomial Square
Factoring a
Factor 4x2+24x+36 Trinomial Square
1. Recognize 1st&Last Terms are Squares
Factor 4x2+24x+36 (r is the sq. root of the first square)
(s is the sq. root of the second square)
r=2x s=6
Factoring a
Factor 4x2+24x+36 Trinomial Square
1. Recognize 1st&Last Terms are Squares
Factor 4x2+24x+36 (r is the sq. root of the first square)
(s is the sq. root of the second square)
r=2x s=6
2rs=2(2x)(6)=24x 2. The middle term must be ±2•r•s
(THIS IS A CHECK IF THE TRINOMIAL IS
MONOMIAL SQUARE)
Factoring a
Factor 4x2+24x+36 Trinomial Square
1. Recognize 1st&Last Terms are Squares
Factor 4x2+24x+36 (r is the sq. root of the first square)
(s is the sq. root of the second square)
r=2x s=6
2rs=2(2x)(6)=24x 2. The middle term must be ±2•r•s
3. Use the Trinomial Square Formula
(2x 6) 2
r 2rs s (r s)
2 2 2
Comparing Difference of Squares and
Trinomial Squares
Factoring a
Factor 16x4-36 Difference of Squares Factor 4x2+24x+36
Factoring a
Trinomial Square
1. Recognize Both Terms are Squares 1. Recognize 1st&Last Terms are Squares
16x 36
4
(r is the sq. root of the first square)
Factor 4x2+24x+36 (r is the sq. root of the first square)
(s is the sq. root of the second square)
r=4x2 s=6 (s is the sq. root of the second square) r=2x s=6
2. One term must be negative for Diff. 2rs=2(2x)(6)=24x 2. The middle term must be ±2•r•s
3. Use the Monomial Square Formula
(4 x 2 6)(4 x 2 6) 3. Use the Diff. Of Squares Formula
(2x 6) 2
r s (r s)(r s)
2 2
r 2 2rs s2 (r s) 2
Factor Completely
50x 8
2 LOOK FOR
ADDITIONAL FACTORS
• (25x 4)
2
2
Factor Completely
50x 8
2 KEEP FACTORING
UNTIL YOU CAN’T
FACTOR ANYMORE
2 • 25x 4
2
2 (5x 2)(5x 2)
FACTOR COMPLETELY
32x 8 2
FACTOR COMPLETELY
32x 8 2
2 (16x 8 1)
FACTOR COMPLETELY
32x16 2
2 (16x 8 1)
(4x 4 1) (4x 4 1)
FACTOR COMPLETELY
32x16 2
2 (16x 8 1)
(4x 4 1) (4x 4 1)
(4 x 4 1) (2x 2 1) (2x 2 1)
2
FACTOR COMPLETELY
GCF TERMS
2 3 4
DOS BOX
X-BOX
Always factor GCF 1st and then factor
and then check for additional factors.
Factoring to Solve Equations
Factoring Polynomial Notes & Tips:
• If a trinomial “X” factor is unsolvable it’s Prime
• The “Box” method does not solve all 4-term polynomials
• Factoring isn’t complete until all polynomials are prime.
Think of a factor tree.
• Make sure 3 & 4 term polynomials are in descending
order. i.e.(change x2+4+3x to x2+3x+4)
• The minus sign in a DOS can be the first monomial square.
i.e.(-x2+36)
Factoring ax2+bx+c
Factor 10x 20x 3 2
GCF Factor “T” Factor 3y2-y-4 Polynomials
-12 1. Set up & Solve “X” factor:
(x 2) 1. Find Greatest Common Number •Put ac on top
2. Find Greatest Common Variable
3 -4 •Put b on bottom
10x 10x 20x
2 3 2
-1
•Determine side factors
2. Set up & Solve “BOX”:
3. Divide GCF into each Term
•Put First Term in First Box
Mental Check 4. Do a Mental Check 3y -4 •Put Last Term in Last Box
Qu i ckTi me ™ a nd a
TIFF (Un co mp res se d) de co mp res so r
a re ne ed ed to se e thi s pi ctu re. •Put Side Factors Times x in Box
y 3y2 -4y •Determine GCF of upper boxes
10x2(x-2)
and top & side factors for ANSWER
1 3y -4
=(y+1)(3y-4)
Factor Polynomials
Factoring 4 Term
x3+3x2+2x+6 Polynomials
x +3 1. Build Box
Put 4 terms in boxes in
x2 x3 3x2 descending
1 2
order.
3 4
+2 2x 6
3. Solve Box
• Determine GCF of upper boxes
=(x+3)(x2+2) • Answer is Top Factors times
Side Factors.
