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 :

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 + 20y^two
– 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(2y^two + 4y – 1)

Check :

Multiply to check your answer.

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

=  10y^three + 20y² – 5y

The production is the original polynomial.

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 – 8a^two

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(y^two + 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)

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 – 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.

=  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(3x^two) – 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.

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.

=  3x^two(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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