How to find the gcf for algebraic expressions :

To find the greatest common factor of the given algebraic expressions we have to follow the steps:

**Step 1:**

For algebraic expression we have to find factors of them using algebraic identities or factoring polynomials.

**Step 2:**

List the common factors.

(a + b)² = a² + 2 ab + b²

(a - b)² = a² - 2 ab + b²

a² - b² = (a + b) (a - b)

a³ + b³ = (a + b)(a²- ab + b²)

a³- b³= (a - b)(a² + ab + b²)

(a + b + c)²= a² + b²+ c² + 2ab + 2bc + 2ca

(a + b)³=a³ + 3 a²b + 3 ab² + b³

(a - b)³=a³- 3 a²b + 3 ab² - b³

**Question 1 :**

Find the GCD of the following c² - d² , c (c - d)

**Solution :**

**Step 1 :**

c² - d² = (c + d) (c - d)

by comparing c²-d² with the algebraic identity (a²-b²) =(a+b)(a-b), we get the factors (c + d) (c -d)

c (c - d) = c (c - d)

**Step 2 :**

Common factor is (c -d)

Hence, the greatest common factor is (c -d)

Let us see the next example on "How to find the gcf for algebraic expressions".

Question 2 :

Find the GCD of the following x⁴ - 27 a³ x , (x - 3a)²

**Solution:**

**Step 1 :**

x⁴ - 27 a³ x = x (x³ - 27 a³)

(x - 3a)² = (x - 3a) (x - 3a)

**Step 2 :**

The common factor is (x -3a)

Hence, the greatest common factor is (x - 3a)

Let us see the next example on "How to find the gcf for algebraic expressions".

Question 3 :

Find the GCD of the following m² - 3 m - 18 , m² + 5 m + 6

**Solution:**

**Step 1 :**

m² - 3 m - 18 = (m - 6) (m + 3)

m² + 5 m + 6 = (m + 2) (m + 3)

**Step 2 :**

The common factor is (m + 3)

Hence, the greatest common factor is (m + 3)

Let us see the next example on "How to find the gcf for algebraic expressions".

Question 4 :

Find the greatest common factor of a^(m + 1), a^(m + 2) , a^(m + 3)

**Solution:**

**Step 1 :**

a^(m + 1) = a^m **x **a

a^(m + 2) = a^m** x** a²

a^(m + 3) = a^m **x **a³

The common factors are a^m **x **a

**Step 3 :**

Multiply the common factors a^m and a

Hence, the greatest common factor is a ^(m + 1)

Let us see the next example on "How to find the gcf for algebraic expressions".

Question 5 :

Find the greatest common factor of x² y + x y² , x² + x y

**Solution:**

**Step 1 :**

x² y + x y² = x y (x + y)

x² + x y = x (x + y)

**Step 2 :**

The common factor is (x + y)

Hence, the greatest common factor is (x + y)

Let us see the next example on "How to find the gcf for algebraic expressions".

Question 6 :

Find the greatest common factor of 3 (a- 1) , 2 (a - 1)² , (a² - 1)

**Solution :**

**Step 1 :**

3 (a- 1) = 3 (a - 1) (cannot be factored here after)

2 (a - 1)² = 2 (a - 1) (a - 1)

(a² - 1) = (a + 1) (a - 1)

**Step 2 :**

The common factor is (a - 1)

Hence, the greatest common factor is (a - 1)

Let us see the next example on "How to find the gcf for algebraic expressions".

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