# 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 :**

4x^{two}

– 3x

**Solution :**

Find the GCF :

4x^{2} = two

⋅ 2

⋅

x

⋅

ten

3x = 3⋅

x

The GCF of 4x^{2}

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)

=

4x^{2} – 3x

The product is the original polynomial.

**Case ii :**

10y^{3} + 20y^{two}

– 5y

**Solution :**

Discover the GCF :

10y^{iii}

= 2

⋅

5

⋅

y

⋅ y

⋅ y

20y^{2}

= 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.

10y^{3} + 20y^{2} – 5y = 2y^{2}(5y) + 4y(5y) – ane(5y)

Employ the Distributive Property to factor out the GCF.

=5y(2y^{two} + 4y – 1)

Check :

Multiply to check your answer.

5y(2y^{ii} + 4y – 1) =5y(2y^{2}) + 5y(4y) + 5y(-1)

= 10y^{three} + 20y^{2} – 5y

The production is the original polynomial.

**Example 3 :**

-12a – 8a^{ii}

**Solution :**

Both coefficients are negative. Factor out -1.

-12a – 8a

^{2
}=-i(12a + 8a

^{2})

Find the GCF :

12a =

2

⋅

2

⋅

iii

⋅

a

8a^{2} =

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 – 8a^{two}

The product is the original polynomial.

**Example four :**

5x^{2}

+ 7

**Solution :**

Discover the GCF :

5x^{2}

=

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(y^{2} + 5) + (y^{2} + five)

**Solution :**

(y

^{2}

+ five) is a common binomial factor.

= -y(y^{2} + 5)

+

(y^{ii} + five)

= -y(y^{2} + 5) + 1(y^{two} + 5)

Factor out (y

^{2}

+ 5).

=(

y^{2} + 5

)(-y + ane)

=(

y^{2} + 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)

Factor out (n + 4).

=(n + four)(9n – 5)

**Example 8 :**

-3y^{2}(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 :**

12x^{3} – 9x^{two}

+ 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.

=

3x^{two}

(4x – iii) +

5(4x – iii)

(4x – 3) is a common factor.

= 3x^{2}

(4x – 3)

+ 5(4x – 3)

Factor out (4x – three).

=

(4x – 3)(

3x^{ii} + 5)

Check :

Multiply to check your respond.

(4x – 3)(

3x^{ii} + 5) = 4x(3x^{2}) + 4x(5) – 3(3x^{two}) – 3(5)

= 12x^{3} + 20x – 9x^{ii} – 15

=

12x^{3} – 9x^{2} + 20x – xv

The product is the original polynomial.

**Case 10 :**

9a^{3} + 18a^{2} + a + ii

**Solution :**

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

= (9a^{3} + 18a^{2}) + (a + 2)

Gene out the GCF of each grouping.

=

9a

^{2}

(a + 2) +

1(a + two)

(a + 2) is a mutual gene.

= 9a^{2}

(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(9a^{2}) + a(1) + two(9a^{ii}) + 2(1)

= 9a^{3} + a + 18a^{2} + two

= 9a^{3} + 18a^{2} + 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).

=

-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 3x^{3}

– 15x^{2}

+ 10 – 2x past group.

**Solution :**

= 3x^{3} – 15x^{2} + ten – 2x

Group terms.

= (3x^{3} – 15x^{ii}) + (10 – 2x)

Factor out the GCF of each group.

= 3x^{two}(ten

– v

) + 2(five – 10)

Write (five – 10) as -1(10 – 5).

= 3x^{2}(10 – five) + 2(-i)(10 – five)

= 3x^{2}(x – 5) – 2(x – v)

(x – 5) is a common cistron.

= 3x^{2}

(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 (7x^{2}

+ x) cm^{2}. Factor this polynomial to find possible expressions for the dimensions of the solar console.

**Solution :**

A =7x

^{2}

+ 10

The GCF of 7x^{2}

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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