8 factoring out gcf

Factoring Out GCF

To factor means to rewrite a quantity as a product (in a nontrivial way).

Factoring Out GCF

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime.

Factoring Out GCF

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example A. Factor 12 completely.

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime, incomplete

factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are 3,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are 3, x,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are 3, x, y2,

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..

A common factor of two or more quantities is a factor belongs to all the quantities.

Example A. Factor 12 completely.

12 = 3 * 4 = 3 * 2 * 2

not prime factored completely

To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime.

Factoring Out GCF

Example B.

a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.b. The common factor of 4ab, 6a are 2, a, 2a.c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..d. The common factor of a(x+y), b(x+y) is (x+y).

A common factor of two or more quantities is a factor belongs to all the quantities.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36}

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a}

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2}

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.d. GCF{x3y5, x4y6, x5y4} =

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.d. GCF{x3y5, x4y6, x5y4} = x3y4.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.d. GCF{x3y5, x4y6, x5y4} = x3y4.

The Extraction LawDistributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences.

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.d. GCF{x3y5, x4y6, x5y4} = x3y4.

The Extraction LawDistributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences.

AB ± AC A(B±C)

The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.

Factoring Out GCF

Example C. Find the GCF of the given quantities.

a. GCF{24, 36} = 12.b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.d. GCF{x3y5, x4y6, x5y4} = x3y4.

The Extraction LawDistributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences.

AB ± AC A(B±C)

This procedure is also called “factoring out a common factor”. To factor, the first step always is to factor out the GCF.

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y

(the GCF is y)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y

(the GCF is y)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

(the GCF is y)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a

(the GCF is y)

(the GCF is 2a)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a

(the GCF is y)

(the GCF is 2a)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3)

(the GCF is y)

(the GCF is 2a)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

(the GCF is y)

(the GCF is 2a)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Factoring Out GCFExample D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y) Pull out the common factor (x + y)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4)

b. Factor (2x – 3)3x – 2(2x – 3)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4)

b. Factor (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3),

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCF

We may pull out common factors that are ( )'s.

Example E. Factor

a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4)

b. Factor (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)

(the GCF is y)

(the GCF is 2a)

(the GCF is 6x2y2)

Example D. Factor out the GCF.a. xy – 4y = y(x – 4)

b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

Factoring Out GCFWe sometimes pull out a negative sign from an expression.

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

We sometimes pull out a negative sign from an expression.

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

We sometimes pull out a negative sign from an expression.

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5)

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1)

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1)

b. 3x – 3y + ax – ay

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

We sometimes pull out a negative sign from an expression.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1)

b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay)

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1)

b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y)

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

We sometimes pull out a negative sign from an expression.

Factoring Out GCF

Example F. Pull out the negative sign from –2x + 5.

–2x + 5 = –(2x – 5)

Sometime we have to use factor twice.

Example G. Factor by pulling out twice.

a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1)

b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y)

We sometimes pull out a negative sign from an expression.