Module Three:

Distributive Property and Factoring

 

Distributive Property

 

The distributive property involves multiplication along with addition and/or subtraction and is stated as follows:

a(b + c) = ab + ac

or

a(b – c) = ab – ac

Example 1.   Simplify:       5x(4x – 9)

 

Step 1.           

Step 2.            20x2 – 45x

 

 

Example 2.  

                        Simplify:       3(2x + y – 4)

 

Step 1.           

Step 2.            6x + 3y – 12

 

 

Example 3.  

                        Simplify:       (x – 3)(4x + 5)

 

Step 1.            x(4x + 5) – 3(4x + 5)

Step 2.            4x2 + 5x – 12x – 15

Step 3.            4x2 – 7x – 15

 

Example 4.  

                        Simplify:       (2x – 1)(3x – 4)

 

Step 1.            2x(3x – 4) – 1(3x – 4)

Step 2.            6x2 – 8x – 3x + 4

Step 3.            6x2 – 11x + 4

 

Example 5.  

                        Simplify:       (3a – 7)(a + c)

 

Step 1.            3a(a + c) – 7(a + c)  

Step 2.            3a2 + 3ac – 7a – 7c

or                     3a2 + (3c – 7)a – 7c

 

Example 6.  

                        Simplify:       (5x + 2y)2

 

Step 1.            (5x + 2y)(5x + 2y)      

Step 2.            5x(5x + 2y) + 2y(5x + 2y)

Step 3.            25x2 + 10xy + 10xy + 4y2

Step 4.            25x2 + 20xy + 4y2

or                     25x2 + 4y2 + 20xy

 

The following example is occasionally confused with the distributive property.

 

Example 7.  

                        Simplify:       (5x2y)3

 

Step 1.            (5x2y) (5x2y) (5x2y)

Step 2.            .

Step 3.            125x6y3

 

Note that there is no addition or subtraction indicated within the parentheses, so the distributive property does not apply. 

 

Factoring

 

For algebraic expressions where the distributive property is used in reverse, or undone, the resulting procedure is called factoring.

 

Example 8.  

                        Factor:          20x2 – 45x

 

                        Using the result from Example 2, we will factor this expression showing that factoring means to express a quantity as a product of factors.

 

Step 1.            The greatest common factor(GCF) is 5x.

Step 2.            Rewrite the expression using the GCF

                                   

Step 3.            Applying the distributive property in reverse we obtain

                                    5x(4x – 9)

 

Example 9.  

                        Factor:          8y3 + 20y2 – 28y

 

Step 1.            The GCF is 4y

Step 2.            4y · 2y2 + 4y · 5y – 4y · 7

Step 3.            4y(2y2 + 5y – 7)

 

 

Example 10.

                        Factor:          4x2 – 7x – 15

                        Note: refer to Example 3

 

Step 1.            Set up 2 sets of parentheses

                                    (     )(     )

Step 2.            The factors of 4x2 are x, 4x or 2x,2x

                        The factors of – 15 are:  

 

Factors

 

Factors

– 1

15

 

1

– 15

3

– 5

 

– 3

5

 

Step 3.            Trying combinations of these factors in the parentheses results in the correct factors (x – 3)(4x + 5)

 

 

 

Example 11.

                        Factor:          x2 – 5x – 24

                       

Step 1.            (     )(     )

Step 2.            The factors of x2 are x, x

                        The factors of – 24 are:         

 

Factors

 

Factors

–1

24

 

1

– 24

2

– 12

 

– 2

12

– 3

8

 

3

– 8

4

– 6

 

– 4

6

           

Step 3.            (x – 8)(x + 3)

 

 

 

Example 12.

                        Factor:          16x2 – 40x + 25

 

Step 1.            (     )(     )

Step 2.            (4x – 5)(4x – 5)

Step 3.            (4x – 5)2

 

 

In the preceding three examples, we factored trinomial expressions.  In the next two examples, we review the method of factoring the difference of two squares.

 

Example 13.

                        Factor:          4x2 – 49

 

Step 1.            (    +    )(    –    )

Step 2.            (2x +   )(2x –  )

Step 3.            (2x + 7)(2x – 7)

 

 

Example 14.

                        Factor:         

 

Step 1.            (    +    )(   –    )

Step 2.           

Step 3.           

 

We will now review an example of an equation where one side is in factored form and the other side is equal to zero.

 

Example 15.

                        Is 0 a solution to either of the following equations?

A.                 2y(y – 7) = 0

B.        (y + 3)(y – 5) = 0

 

Recall that the product of two quantities equals zero if and only if at least one of the factors is zero.

 

For equation A,         2y(y – 7) = 0,

both 2y and y – 7 are factors.  Set each factor equal to 0 and solve for y.

            If y – 7 = 0 then y = 7.

            If 2y = 0 then y = 0.

0 is a solution for equation A.

 

For equation B,         (y + 3)(y – 5) = 0,

            both y + 3 and y – 5 are factors.

            If y + 3 = 0 then y = – 3.

            If y – 5 = 0 then y = 5.

            0 is not a solution for equation B.

 

Practice Problems

 

 

1.   Simplify:   – 6(z – 2k + 3)

 

Solution to problem 1:

 

– 6z + 12k – 18

 

 

2. Simplify:   8x( 4x5– 3x2)

 

Solution to problem 2:

 

32x6 – 24x3

 

 

3. Simplify:    (8p + q)2

 

Solution to problem 3:

 

64p2 + 16pq + q2

 

 

4. Simplify:    (– 4a2c3)2

 

Solution to problem 4:

 

16a4c6

 

 

5. Simplify:     (3d + e)(2x – 5)

 

Solution to problem 5:

 

6dx – 15d + 2xe – 5e

 

 

6.  Factor:   3y9 – 6y8 – 27y4

 

Solution to problem 6:

 

 

            GCF =  3y4                            

      

             3y4(y5 – 2y4 – 9)

 

 

7. Factor:   x2 – 9xy + 20y2

 

Solution to problem 7:

 

(x – 5y)(x – 4y)

 

 

8. Factor:   81 – 25m2

 

Solution to problem 8:

 

Difference of two squares

(9 + 5m)(9 – 5m)

 

 

 

 

Solution to problem 9:

 

Difference of two squares