FACTORING ALGEBRAIC EXPRESSIONS

Factor each of the following. 

Example 1 :

4x + 24y

Solution :

Greatest common divisor of 4x and 24y is 4.  

Divide 4x and 24y by 4. The results are x and 6y. 

4x + 24y = 4(x + 6y)

Example 2 :

2x² – 8x

Solution :

GCD of 2x² and 8x is 2x.  

Divide 2x² and 8x by 2x. The results x and 4. 

2x² – 8x = 2x(x - 4)

Example 3 :

x² – 3x

Solution :

GCD of x² and 3x is x.  

Divide x² and 3x by x. The results x and 3. 

x² – 3x = x(x - 3)

Example 4 :

3a + 6ab + 9a²

Solution :

GCD of 3a, 6ab and 9a² is 1, 2b and 3a.

3a + 6ab + 9a² = 3a(1 + 2b + 3a)

Example 5 :

ab + bc - 2b

Solution : 

GCD of ab, bc and 2b is b. 

Divide ab, bc and 2b by b. The results are a, c and -2.

ab + bc - 2b = b(a + c – 2)

Example 6 :

3(x – 6) + d(x – 6)

Solution :

GCD of 3(x – 6) and d(x – 6) is (x - 6). 

Divide 3(x – 6) and d(x – 6) by (x - 6). The results are 3 and d. 

3(x – 6) + d(x – 6) = (x – 6)(3 + d)

Example 7 :

2x(x + 4) + 3x + 12

Solution :

In the expression above, GCD of 3x and 12 is 3. 

Divide 3x and 12 by 3. The results are x and 4.

2x(x + 4) + 3x + 12 = 2x(x + 4) + 3(x + 4)

GCD of 2x(x + 4) and 3(x + 4) is (x + 4). 

Divide 2x(x + 4) and 3(x + 4) by (x + 4). The results are 2x and 3.

2x(x + 4) + 3x + 12 = (x + 4)(2x + 3)

Example 8 :

4x²y³ + 6x³y²

Solution :

GCD of 4x²y³ + 6x³y² is 2x²y². 

Divide 4x²y³ and 6x³y² is 2x²y². The results are 2y and 3x.

4x²y³ + 6x³y² = 2xy(2x + 3y)

Example 9 :

x² + 5x + 6

Solution : 

Find two factors of +6 such that the product is +6 and sum is +5. 

Then the two factors of +6 are +2 and +3.

Split 5x into two terms using the factors +2 and +3.

x² + 5x + 6 = x² + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

Example 10 :

x² - 15x + 56

Solution : 

Find two factors of +56 such that the product is 56 and sum is -15. 

Then the two factors of -15 are -7 and -8.

Split -15x into two terms using the factors -7 and -8.

x² - 15x + 56 = x² - 7x - 8x + 56

= x(x - 7) - 8(x - 7)

= (x - 7)(x - 8)

Example 11 :

x² + 2x - 15

Solution : 

Find two factors of -15 such that the product is -15 and sum is +2. 

Then the two factors of -15 are -3 and +5.

Split -2x into two terms using the factors -3 and +5.

x² + 2x - 15 = x² - 3x + 5x - 15

= x(x - 3) + 5(x - 3)

= (x - 3)(x + 5)

Example 12 :

x² - x - 20

Solution : 

Find two factors of -20 such that the product is -20 and sum is -1. 

Then the two factors of -20 are -5 and +4.

Split -x into two terms using the factors -5 and +4.

x² - x - 20 = x² - 5x + 4x - 20

= x(x - 5) + 4(x - 5)

= (x - 5)(x + 4)

Example 13 :

2x² + 9x + 9

Solution : 

The product of 2 and 9 is +18.

Find two factors of +18 such that the product is +18 and sum is +9. 

Then the two factors of +18 are +3 and +6.

Split 9x into two terms using the factors +3 and +6.

2x² + 9x + 9 = 2x² + 3x + 6x + 9

= x(2x + 3) + 3(2x + 3)

= (2x + 3)(x + 3)

Example 14 :

3x² + x - 4

Solution : 

The product of 3 and -4 is -12.

Find two factors of -12 such that the product is -12 and sum is +1. 

Then the two factors of -12 are -3 and +4.

Split +x into two terms using the factors -3 and +4.

3x² + x - 4 = 3x² - 3x + 4x - 4

= 3x(x - 1) + 4(x - 1)

= (x - 1)(3x + 4)

Example 15 :

x² + 8x + 16

Solution : 

x² + 8x + 16 = x² + 2(x)(4) + 4²

= (x + 4)²

= (x + 4)(x + 4)

Example 16 :

x² + 6xy + 9y²

Solution : 

x² + 6xy + 9y² = x² + 2(x)(3y) + (3y)²

= (x + 3y)²

= (x + 3y)(x + 3y)

Example 17 :

4a² - 20ab + 25b²

Solution : 

4a² - 20ab + 25b² = 2²a² - 20ab + 5²b²

= (2a)² - 20ab + (5b)²

= (2a)² - 2(2a)(5b) + (5b)²

= (2a - 5b)²

= (2a - 5b)(2a - 5b)

Problem 18 :

9m² - 16n²

Solution : 

9m² - 16y² = 3²m² - 4²n²

= (3m)² - (4n²)

= (3m + 4n)(3m - 4n)

Problem 19 :

a²b² - c²d²

Solution : 

a²b² - c²d² = (ab)² - (cd)²

= (ab + cd)(ab - cd)

Problem 20 :

x⁴ - y⁴

Solution : 

x⁴ - y⁴ = (x²)² - (y²)²

Let a = x² and b = y².

= a² - b²

= (a + b)(a - b)

Substitute a = x² and b = y².

= (x² + y²)(x² - y²)

=  (x² + y²)(x + y)(x - y)

Example 21 :

x³ + 3x² + 6x + 18

Solution :

GCD of x³ and 3x² is x². 

Divide x³ and 3x² by x². The results are x and 3.

Similarly GCD of 6x and 18 is 6 

Divide 6x and 18 by 6. The results are x and 3.

x³ + 3x² + 6x + 18 = x²(x + 3) + 6(x + 3)

= (x + 3)(x² + 6)

Example 22 :

x³ + 2x² - 9x - 18

Solution :

GCD of x³ and 2x² is x². 

Divide x³ and 2x² by x². The results are x and 2.

Similarly GCD of -9x and -18 is -9 

Divide -9x and -18 by -9. The results are x and 2.

x³ + 2x² - 9x - 18 = x²(x + 2) - 9(x + 2)

= (x + 2)(x² - 9)

= (x + 2)(x² - 3²)

= (x + 2)(x + 3)(x - 3)

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