What is the Greatest Common Factor of 12a and 9a2

What is the Greatest Common Factor of 12a and 9a2

FACTORING Past GCF




Recollect that the Distributive Property states that

ab + ac  =  a(b + c)

The Distributive Holding allows you to factor out the GCF of the terms in a polynomial to write a factored form of the polynomial.

A polynomial is in its factored class when it is written equally a product of monomials and polynomials that cannot be factored further. The expression 2(3x – 4x) is not fully factored considering the terms in the parentheses have a common factor of x.

Factoring by Using the GCF

Factor each polynomial. Check your reply.


Example 1 :

4xtwo
– 3x


Solution :

Find the GCF :

4x2  =  two
⋅ 2


x

ten

3x  =  3

x


The GCF of 4x2
and 3x is x.

Write terms as products using the GCF as
a factor.

=  4x(x) – 3(x)

Employ the Distributive Property to factor out the GCF.

=
ten(4x

 – three
)

Check :


Multiply to check your answer.


10(4x – 3)  =  x(4x) – x(three)


=
4x2 – 3x

The product is the original polynomial.


Case ii :

10y3 + 20ytwo
– 5y


Solution :

Discover the GCF :

10yiii
  =  2

5


 y

⋅ y
⋅ y

20y2
  =  2
⋅ 2



v


 y

⋅ y


5y
  =


5


 y


The GCF of

10y
3
, 20y
2
 and 5y is 5y.

Write terms as products using the GCF as
a cistron.

10y3 + 20y2 – 5y  =  2y2(5y) + 4y(5y) – ane(5y)

Employ the Distributive Property to factor out the GCF.

=5y(2ytwo + 4y – 1)

Check :

Multiply to check your answer.

5y(2yii + 4y – 1)  =5y(2y2) + 5y(4y) + 5y(-1)

=  10ythree + 20y2 – 5y

The production is the original polynomial.

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Example 3 :

-12a – 8aii


Solution :

Both coefficients are negative. Factor out -1.

-12a – 8a
2
=-i(12a + 8a
2)

Find the GCF :

12a  =
2


2



 iii


a

8a2  =
2


2

⋅ 2


 a

⋅ a


The GCF of

12a and 8a
2
is 4a.

Write terms as products using the GCF as
a factor.

=  -ane[3(4a) + 2a(4a)]

Use the Distributive Property to factor out the GCF.

=  -1[4a(three + 2a)]

=  -4a(3 + 2a)

Check :

Multiply to check your answer.

-4a(3 + 2a)  =  -4a(three) – 4a(2a)

=  -12a – 8atwo

The product is the original polynomial.


Example four :

5x2
+ 7


Solution :

Discover the GCF :

5x2
  =


5

 x

⋅ x

seven  =  seven

At that place are no mutual factors other than one.

The polynomial cannot be factored.

Sometimes the GCF of terms is a binomial. This GCF is called a common binomial factor. You factor out a common binomial factor the same fashion you lot cistron out a monomial gene.

Factoring Out a Common Binomial Gene

Factor each expression.


Example v :

vii(x – 3) – 2x(10 – 3)


Solution :

(x – three) is a common binomial gene.

=  7(x – 3)
– 2x(x – three)

Cistron out (x – iii).

=
(x – iii)(7 – 2x)


Example 6 :

-y(y2 + 5) + (y2 + five)


Solution :

(y
2
 + five) is a common binomial factor.

=  -y(y2 + 5)
+
(yii + five)


=  -y(y2 + 5) + 1(ytwo + 5)

Factor out (y
2
 + 5).

=(
y2 + 5
)(-y + ane)

=(
y2 + 5
)(1 – y)


Example 7 :

9n(n + iv) – v(4 + n)


Solution :

Addition is always commutative. So,

4 + due north  =  n + 4

So,

=  9n(due north + 4) – five(n + 4)

(n + four) is a mutual binomial factor.

=  9n(n + 4)
– five(n + 4)

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Factor out (n + 4).

=(n + four)(9n – 5)


Example 8 :

-3y2(y + 2) + 4(y – vii)


Solution :

There are no common factors.

The expression cannot be factored.

Y’all may be able to factor a polynomial by grouping. When a polynomial has iv terms, y’all can make two groups and gene out the GCF from each grouping.

Factoring by Group

Factor each polynomial by grouping. Check your reply.


Example 9 :

12x3 – 9xtwo
+ 20x – fifteen


Solution :

Grouping terms that have a common number or variable every bit a factor.

=  (12x
3
 – 9x
2) + (20x – 15)

Cistron out the GCF of each group.

=
3xtwo
(4x – iii) +
5(4x – iii)

(4x – 3) is a common factor.


=  3x2
(4x – 3)
+ 5(4x – 3)

Factor out (4x – three).



=
(4x – 3)(


3xii + 5)

Check :

Multiply to check your respond.


(4x – 3)(
3xii + 5)  =  4x(3x2) + 4x(5) – 3(3xtwo) – 3(5)


=  12x3 + 20x – 9xii – 15


=
12x3 – 9x2 + 20x – xv

The product is the original polynomial.


Case 10 :

9a3 + 18a2 + a + ii


Solution :

Group terms that take a common number or variable every bit a factor.

=  (9a3 + 18a2) + (a + 2)

Gene out the GCF of each grouping.

=
9a

2
(a + 2) +
1(a + two)

(a + 2) is a mutual gene.

=  9a2
(a + 2) + 1(a + 2)

Gene out (a + 2).


=(a + 2)(9a

2 + ane)

Bank check :

Multiply to bank check your answer.


(a + 2)
(9a

two + one)  =  a(9a2) + a(1) + two(9aii) + 2(1)

=  9a3 + a + 18a2 + two

=  9a3 + 18a2 + a + 2

The product is the original polynomial.

Recognizing opposite binomials can help you factor polynomials. The binomials (5 – x) and (x – 5) are opposites. Notice (v – x) can be written as -one(x – 5).

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=
-1(x – v)

Distributive Property.

=
(-1)(x) +(-i)(-5)

Simplify.

=  -10 + v

Commutative Holding of Improver.

=  5 – x

So,

5 – x  =  -1(ten – five)

Factoring with Opposites


Instance 11 :

Factor 3x3
– 15x2
+ 10 – 2x past group.


Solution :

=  3x3 – 15x2 + ten – 2x

Group terms.

=  (3x3 – 15xii) + (10 – 2x)

Factor out the GCF of each group.

=  3xtwo(ten
 – v
) + 2(five – 10)

Write (five – 10) as -1(10 – 5).

=  3x2(10 – five) + 2(-i)(10 – five)

=  3x2(x – 5) – 2(x – v)

(x – 5) is a common cistron.

=  3x2
(x – 5)
– 2(x – v)

Factor out (10 – 5).



=
(x – v)(3x


2 – 2)



Scientific discipline Awarding


Case 12 :

Lily’southward calculator is powered by solar energy. The area of the solar panel is (7x2
+ x) cm2. Factor this polynomial to find possible expressions for the dimensions of the solar console.


Solution :

A  =7x
2
 + 10

The GCF of 7x2
and ten is ten.

Write each term as a product using the GCF as a factor.

=  7x(x) + ane(x)

Use the Distributive Property to factor out the GCF.

=  10(7x + 1)

Possible expressions for the dimensions of the solar panel are 10 cm and (7x + 1) cm.

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What is the Greatest Common Factor of 12a and 9a2

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