Key Concepts

• Factor expressions
• Factor expressions with negative coefficients
• Factor three- term expressions

4.5 Factor expressions 

• An expression is in factored form only if the entire expression is an indicated product (rewriting as the product of factors).  

• Factoring is a process that changes a sum or difference of terms to a product of factors. 

• A prime expression cannot be factored.  
• The greatest common factor is the greatest factor common to all terms. 

4.5.1 Factor expressions 

Example1: 

Use factoring to write an expression for the length of the pool with the given width. 

4x+20. 

Solution: 

One way: 

Use an area model to represent the area of the swimming pool. 

So, one possible set of dimensions of the length of the pool is x+5 meters long, and the width is 4 meters. 

Another way: 

Use a common factor and the distributive property to factor the expression. 

4x+20 

4x+20. 

So, the pool is (x+5) meters long and 4 meters wide. 

4.5.2 Factor expressions with negative coefficients 

What is meant by coefficients? 

Coefficients are numbers that are multiplied by variables. 

What is meant by negative coefficients? 

Negative coefficients are simply coefficients that are negative numbers. 

Example1: 

Show two different ways to factor -4x – 28. 

Solution: 

One way: 

Use a positive common factor 4 to factor the expression. 

4 is a common factor of -4x and 28. 

4(-x – 7) 

=-4x – 28. 

Another way: 

Use a negative common factor -4 to factor the expression. 

-4 is a common factor of -4x and -28. 

-4(x+7) 

=-4x – 28 

4(-x – 7) and -4(x+7) are equivalent expressions. 

4.5.3 Factor three-term expressions 

What is meant by a term? 

A term is a single mathematical expression.  

It may be a single number (positive or negative), a single variable (a letter), several variables multiplied but never added or subtracted. 

The example below is a three-term expression. 

Example 1: 

Use the G.C.F to factor the expression 16x-24-32y 

Solution: 

Step 1: Find the G.C.F of 16x, -24 and -32y 

      Factors of 16: 1, 2, 4, 8, 16. 

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. 

Factors of 32: 1, 2, 4, 8, 16, 32. 

The G.C.F is 8. 

Step 2: Use the G.C.F and the distributive property to factor the expression. 

16x-24-32y  

= (8)(2x) – (8) (3) – (8)(4y) 

= 8(2x – 3 – 4y) 

Example 2: 

Use the G.C.F to factor the expression 5t-15-20w. 

Solution: 

Step 1: Find the G.C.F of 5t, -15 and -20w 

      Factors of 5: 1, 5. 

Factors of 15: 1, 3, 5, 15. 

Factors of 20: 1, 2, 4, 5, 10, 20. 

     The G.C.F is 5. 

Step 2: Use the G.C.F and the distributive property to factor the expression. 

5t-15-20w. 

             = (5)(t) – (5) (3) – (5)(4w) 

Exercise:

1. Factor the expression.
   1. 18b+20
   2. 12x+36
2. How can you use the distributive property to factor the expression 4x+10?
3. Show different ways to factor -8x-16- 24y.
4. Use the G.C.F to write the factored form of the expression 15x+20y.
5. Factor the expression 6m+15.
6. Show different ways to factor -7n -70.
7. Find the G.C.F of 4x+16.
8. This model shows the area of the garden. Write two expressions that represent the area.

9. Factor out the Greatest Common Factor.

• a.24x + 40xy              
• b.30xy+ 40xy+55y

10. Use the G.C.F to factor the expression 15 x+ 25 xy+ 50.

Concept Map

What have we learned:

• Understand factors expression.
• Understand how to factor expressions with negative coefficients.
• Identify GCF.
• Identify factored expression.
• Understand how to factor three trems expressions.