### 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:

- Factor the expression.
- 18b+20
- 12x+36

- How can you use the distributive property to factor the expression 4x+10?
- Show different ways to factor -8x-16- 24y.
- Use the G.C.F to write the factored form of the expression 15x+20y.
- Factor the expression 6m+15.
- Show different ways to factor -7n -70.
- Find the G.C.F of 4x+16.
- 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.

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