FACTORING EXPRESSIONS

Factor each of the following. 

Example 1 :

4x + 8

Solution : 

Find the largest common divisors for 4x and 8. 

The largest common divisor for 4x and 8 is 4. 

Divide 4x and 8 by 4

4x/4 = x

8/4 = 2

Write the quotients x and 2 inside the parentheses and multiply by the largest common divisor 4.

4(x + 2)

So, 4x + 8 = 4(x + 2).

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 4 and (x+2). 

Distribute 4 to x and 2. 

4(x + 2) = 4(x) + 4(2

4(x + 2)  =  4x + 8

When we multiply 4 and (x + 2), we get the given expression.

So, 4 and (x + 2) are the factors of 4x + 8.

Example 2 : 

Factor :

16a + 64b - 4c

Solution : 

Find the largest common divisors for 16a, 64b and 4c. 

The largest common divisor for  16a, 64b and 4c is 4. 

Divide 16a, 64b and 4c by 4 

16a/4 = 4a

64b/4 = 16b

4c/4 = c

Write the quotients 4a, 16b and c inside the parentheses and multiply by the largest common divisor 4.

4(4a + 16b - c)

So, 16a + 64b - 4c = 4(4a + 16b - c) 

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 4 and (4a + 16b - c). 

Distribute 4 to 4a, 16b and c. 

4(4a + 16b - c) = 4(4a) + 4(16b) + 4(-c)

= 16a + 64b - 4c

When we multiply 4 and (4a + 16b - c), we get the  given expression.

So, 4 and (4a + 16b - c) are the factors of 16a + 64b - 4c.

Example 3 : 

Factor :

5x² - 15x

Solution : 

Find the largest common divisors for 5x² and 15x. 

The largest common divisor for 5x² and 15x is 5x. 

Divide 5x² and 15x by 5x.

5x²/5x = x

15x/5x = 3

Write the quotients x and 3 inside the parentheses and multiply by the largest common divisor 5x.

5x(x - 3)

So, 5x² + 15x = 5x(x - 3).

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 5x and (x - 3). 

Distribute 5x to x and 3. 

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

5x(x - 3) = 5x² - 15x

When we multiply 5x and (x - 3), we get the given expression.

So, 5x and (x - 3) are the factors of  5x² - 15x.

Example 4 :

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 5 :

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 6 :

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 7 :

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 8 :

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 9 :

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 10 :

x² + 8x + 16

Solution : 

x² + 8x + 16 = x² + 2(x)(4) + 4²

= (x + 4)²

= (x + 4)(x + 4)

Example 11 :

x² + 6xy + 9y²

Solution : 

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

= (x + 3y)²

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

Example 12 :

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 13 :

9m² - 16n²

Solution : 

9m² - 16y² = 3²m² - 4²n²

= (3m)² - (4n²)

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

Problem 14 :

a²b² - c²d²

Solution : 

a²b² - c²d² = (ab)² - (cd)²

= (ab + cd)(ab - cd)

Problem 15 :

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 16 :

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 17 :

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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