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Factoring
Polynomials
Greatest Common Factor and
Factoring by Grouping
Finding GCF (Greatest Common Factor)
the largest natural number that divides all given numbers evenly
Find the GCF of 24 and 60
List Factors
Prime Factorization
24
2
Factors of 24:
12
4
3
2
1 2 3 4 6 8 12 24
2
24=2*2*2*3
60
Factors of 60:
2
1 2 3 4 5 6 10 12 15 20 30 60
30
10
2
3
5
60=2*2*3*5
GCF = 12
GCF = 2*2*3 = 12
Finding GCF Of Variables
Find the GCF of x2y3z and x3z
Consider each variable that exists in each Monomial
X
x2 and x3
GCF of x2 and x3 is x2
y
y3 and y0
GCF of y3 and y0 is y0=1
z
z and z
GCF of z and z is z
Therefore, GCF of x2y3z and x3z is x2*1*z = x2z
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The simplest method of factoring a polynomial is to factor out
the greatest common factor (GCF) of each term.
Example: Factor 18x3 + 60x.
GCF of 18 and 60 = 6
GCF
x3
Determine GCF of the coefficients of each term.
and x = x
Determine GCF of the variables in each term.
The GCF of the polynomial is the product of
the GCF’s found above.
GCF = 6x
18x3 + 60x = 6x 18x3 + 60x
6x
3x2
Factored form
GCF * orig. polynomial
GCF
10
18x3 + 60x = 6x 18x3 + 60x Rewrite and Simplify within the
6x
6x parentheses.
18x3 + 60x = 6x (3x2 + 10)
Check the answer by multiplication.
6x(3x2 + 10) = 6x(3x2) + 6x(10) = 18x3 + 60x
The simplest method of factoring a polynomial is to factor out
the greatest common factor (GCF) of each term.
Example: Factor 24x2y + 60x3.
GCF of 24 and 60 = 12 Determine GCF of the coefficients of each term.
GCF x2y and x3 = x2
Determine GCF of the variables in each term.
The GCF of the polynomial is the product of
the GCF’s found above.
GCF = 12x2
24x2y + 60x3 = 12x2 24x2y + 60x3
Factored form
12x2
2y
GCF * orig. polynomial
GCF
5x
24x2y + 60x3 = 12x2 24x2y + 60x3 Rewrite and Simplify within the
12x2
12x2 parentheses.
24x2y + 60x3 = 12x2(2y + 5x)
Check the answer by multiplication.
12x2(2y + 5x) = 12x2 (2y) + 12x2 (5x) = 24x2y + 60x3
The simplest method of factoring a polynomial is to factor out
the greatest common factor (GCF) of each term.
Example: Factor 4x2 -12x+ 20.
GCF of 4, 12 and 20 = 4 Determine GCF of the coefficients of each term.
GCF x2, x , 0 = N/A
Determine GCF of the variables in each term.
The GCF of the polynomial is the product of
the GCF’s found above.
GCF = 4
4x2
4x2
-12x +20 = 4 4x2 -12x +20
4
-12x +20 = 4
x2
4x2
4
3x
Factored form
GCF * orig. polynomial
GCF
5
-12x +20
4
4
Simplify.
4x2 -12x +20 = 4(x2 -3x +5)
Check the answer by multiplication.
4(x2 – 3x + 5) = 4x2 – 12x + 20
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A common binomial factor can be factored out of certain
expressions.
Example: Factor the expression 5(x + 1) – y(x + 1).
5(x + 1) – y(x + 1) = (5 – y)(x + 1)
(5 – y)(x + 1) = 5(x + 1) – y(x + 1) Check.
Some polynomials can be factored by grouping terms to produce
a common binomial factor.
Examples: 1. Factor xy + 2y + 7x + 14.
xy + 2y + 7x + 14. = (xy + 2y) + (7x + 14). Group terms.
= y(x + 2) + 7(x + 2)
Factor each pair of terms.
= (x + 2)( y + 7)
Factor out the common
binomial.
2. Factor 6x3 - 8x2 + 3xy - 4y.
6x3 - 8x2 + 3xy - 4y.
= (6x3 - 8x2)+ (3xy - 4y).
= 2x2(3x – 4) + y(3x – 4)
Group terms.
Factor.
= (2x2 + y)(3x – 4)
Some polynomials can be factored by grouping terms to produce
a common binomial factor.
Examples: 3. Factor 2xy + 3y – 4x – 6.
2xy + 3y – 4x – 6 = (2xy + 3y) – (4x + 6) Group terms.
= (2x + 3)y – (2x + 3)2 Factor each pair of terms.
= (2x + 3)( y – 2)
Factor out the common
binomial.
4. Factor 2a2 + 3bc – 2ab – 3ac.
2a2 + 3bc – 2ab – 3ac = 2a2 – 2ab – 3ac + 3abc Rearrange terms.
= (2a2 – 2ab) – (3ac + 3bc) Group terms.
Factor.
= 2a(a – b) – 3c(a – b)
= 2a(a – b) – 3c(a – b)
= (2a – 3c)(a – b)
Factor.
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