Download Factoring a Monomial from a Polynomial

MTH 10905
Algebra
Factoring a Monomial
from a Polynomial
Chapter 5 Section 1
Identify Factors
 Factor an expression means to write the
expression as a product of its factors
 Factoring can be used to solve equations
and perform operations on fractions.
 Factoring is the reverse process of
multiplying.
Identify Factors
 Remember:
 A term is parts that are added
 For example:
2x – 3y – 5
2x + (-3y) + (-5)
 A factor is variables that are multiplied
 Therefore, if a • b = c then a and b are
factors of c.
Identify Factors
 Example:
3 • 5 = 15
3 and 5 are factors of 15
 Example:
x3 • x4 = x7
x3 and x4 are factors of x7
We general list only the positive factors, however,
the negatives or opposites of each of these are also
factors.
Identify Factors
 Example:
x(x+2) = x2 + 2x
x and (x + 2) are factors of x2 + 2x
 Example:
(x – 1)(x + 3) = x2 + 2x -3
(x – 1) and (x + 3) are factors of x2 + 2x -3
Identify Factors
Example:
List the factors of 9x3
1 • 9x3
3 • 3x3
9 • x3
x • 9x2
3x • 3x2
9x • x2
Therefore: 1, 3, 9, x, 3x, 9x, x2, 3x2, 9x2, x3, 3x3, 9x3
and the opposites of these are factors of 9x3
Examples of
Multiplying and Factoring
Example: Multiply
7(x + 2)
(7)(x) + (7)(2)
7x + 14
Example: Factoring
7x + 14
7(x + 2)
Examples of
Multiplying and Factoring
Example: Multiply
2(x – 2)(3x + 1)
2[(x)(3x)+(x)(1)+(-2)(3x)+(-2)(1)]
(2)(x)(3x)+(2)(x)(1)+(2)(-2)(3x)+(2)(-2)(1)
6x1+1 + 2x – 12x – 4
6x2 – 10x – 4
Example: Factoring
6x2 – 10x – 4
2(x – 2)(3x + 1)
Examples of
Multiplying and Factoring
Example: Multiply
(x – 5)(x – 4)
(x)(x) + (x)(-4) + (-5)(x) + (-5)(-4)
x1+1 – 4x – 5x + 20
x2 – 9x + 20
Example: Factoring
x2 – 9x + 20
(x – 5)(x – 4)
Determine the GCF
of Two or More Numbers
To factor we need to make use the Greatest Common
Factor (GCF).
If you are having trouble seeing the GCF you can start
with a common factor and continuing pulling out the
common factors until no common factors remain.
Remember that the GCF of two or more numbers is
the greatest number that divides into all the numbers
Example:
GCF of 6 and 8 is 2
Determine the GCF
of Two or More Numbers
When the GCF is not easy to find we can find it by
writing each number as a product of prime numbers.
Prime Number is an integer greater than 1 that has
exactly two factors, itself and one.
The first 15 prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Determine the GCF
of Two or More Numbers
Positive integers greater than 1 that are not prime are
called composite numbers.
The first 15 composite numbers are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25
All even number greater than 2 are composite numbers.
The number 1 is called a unit. One is not a prime
number and it is not a composite number.
Determine the GCF
of Two or More Numbers
Example:
Write 54 as a product of prime numbers.
54 = 2 • 3 • 3 • 3 = 2 • 33
6
2
3
9
3
3
Prime Factorization
of 54
Determine the GCF
of Two or More Numbers
Example:
Write 80 as a product of its prime factors.
80 = 2 • 2 • 2 • 2 • 5 = 24 • 5
8
2
2
10
4
2
2
2
Prime Factorization
of 80
5
2
5
Determine the GCF
of Two or More Numbers
1. Write each number as a product of prime
factors.
2. Determine the prime factors common to all
numbers.
3. Multiply the common factors to get the GCF
Determine the GCF
of Two or More Numbers
Example:
Determine the GCF of 48 and 80.
48
(6)
(8)
(2)(3) (2)(4)
(2)(3) (2)(2)(2)
2•3•2•2•2
24 • 3
80
(8)
(10)
(2)(4) (2)(5)
(2)(2)(2) (2)(5)
2•2•2•2•5
24 • 5
GCF = 24 = 16
Determine the GCF
of Two or More Numbers
Example:
Determine the GCF of 56 and 124.
56
(2) (28)
(2) (2)(14)
(2) (2)(2)(7)
2•2•2•7
23 • 7
124
(2) (62)
(2) (2)(31)
2 • 2 • 31
22 • 31
GCF = 22 = 4
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the terms:
y8, y2, y6, and y10
To determine the GCF of two or more terms, take each factor the
largest number of times that it appears in all the terms.
y8 =
y2 =
y6 =
y10 =
y2 • y2
y2 • 1
y2 • y4
y2 • y8
GCF = y2
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the terms:
a2b7, a4b, and a8b2
a2b7 = a2
• b • b6
a4b = a2 • a2 • b
a8b2 = a2 • a6 • b • b
GCF = a2b
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the terms:
pq, p3q, and q2
pq = p
•q
p3q = p • p 2 • q
q2 =
q•q
GCF = q
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the terms.
-12b3, 18b2, and 28b
-12b3 = -1 • 2 • 2 • 3
•b
• b2
18b2 =
2•
3• 3
•b•b
28b =
2•2
•7•b
GCF = 2b
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the terms.
y3, 9y5, and y2
y3 =
y • y2
9y5 = 9 •
y 2 • y3
y2 =
y2
GCF = y2
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the pair of terms.
y(y - 2) and 3(y – 2)
y(y – 2) =
y • (y – 2)
3(y – 2) = 3 • (y – 2)
GCF = (y – 2)
Determine the GCF
of Two or More Terms
Example:
Determine the GCF of the pair of terms.
3(x + 6) and x + 6
3(x + 6) = 3 • (x + 6)
1(x + 6) = 1 • (x + 6)
GCF = (x + 6)
Factor a Monomial
from a Polynomial
Steps to Factor a Monomial from a Polynomial:
1.
Determine the greatest common factor of all terms in the
polynomial
2.
Write each term as a product of the GCF and its other
factors
3.
Use the distributive property to factor out the GCF
Example: Factor 8y + 12
8y + 12 = (4 • 2y) + (4 • 3)
= 4(2y + 3)
GCF = 2 • 2 = 4
Factor a Monomial
from a Polynomial
Example: Factor
24x – 18
GCF = 6
24x – 18 = (6 • 4x) – (6 • 3)
= 6(4x – 3)
To check the factoring process, multiply the
factors using the distributive property. If the
factoring is correct, the product will be the
polynomial you start with.
Factor a Monomial
from a Polynomial
Example: Factor
8w2 + 12w6
GCF = 2w • 2w = 4w2
8w2 + 12w6 = (4w2 • 2) + (4w2 • 3w4)
= 4w2(2 + 3w4)
Check:
4w2 (2 + 3w4)
(4w2)(2) + (4w2)(3w4)
8w2 + 12w6
Factor a Monomial
from a Polynomial
Example:
Factor
8x5 + 12x2 – 44x
GCF = 2x • 2x = 4x
8x5 + 12x2 – 44x
= (4x • 2x4)+ (4x • 3x) – (4x • 11)
= 4x(2x2 + 3x – 11)
Factor a Monomial
from a Polynomial
Example:
Factor
60p2 – 12p – 18
GCF = 2 • 3 = 6
60p2 – 12p – 18
= (6 • 10p2)– (6 • 2p) – (6 • 3)
= 6(10p2 – 2p – 3)
Factor a Monomial
from a Polynomial
Example:
Factor
3x3 + x2 + 9x2y
GCF = x2
3x3 + x2 + 9x2y
= (x2 • 3x) + (x2 • 1) + (x2 • 9y)
= x2(3x + 1 + 9y)
Factor a Monomial
from a Polynomial
Example:
Factor
x(6x + 5) + 9(6x + 5)
GCF = 6x + 5
x(6x + 5) + 9(6x + 5)
= x • (6x + 5) + 9 • (6x + 5)
= (6x+5)(x + 9)
Factor a Monomial
from a Polynomial
Example:
Factor
3x(x – 3) – 2(x – 3)
GCF = x – 3
3x(x – 3) – 2(x – 3)
= 3x • (x – 3) – 2 • (x – 3)
= (x – 3)(3x –2)
Factor a Monomial
from a Polynomial
Example:
Factor
6y(5y – 2) – 5(5y – 2)
GCF = 5y – 2
6y(5y – 2) – 5(5y – 2)
= 6y • (5y – 2) – 5 • (5y – 2)
= (5y – 2)(6y – 5)
IMPORTANT
Whenever you are factoring a
polynomial by any method; the first
step is to see if there are any
common factors (other than 1) to all
the terms in the polynomial. If yes,
factor the GCF from each term using
the distributive property.
HOMEWORK 5.1
Page 298:
#49, 51, 55, 61, 69, 79, 81, 89, 91