# Factoring Trinomials

Learn how to use FOIL, “Difference of Squares” and “Reverse FOIL” to factor trinomials.

## Step by step guide to Factoring Trinomials

• To factor trinomials sometimes we can use the “FOIL” method (First-Out-In-Last):
$$\color{blue}{(x+a)(x+b)=x^2+(b+a)x+ab}$$
• Difference of Squares”:
$$\color{blue}{ a^2-b^2=(a+b)(a-b)}$$
$$\color{blue}{ a^2+2ab+b^2=(a+b)(a+b)}$$
$$\color{blue}{ a^2-2ab+b^2=(a-b)(a – b)}$$
• Reverse FOIL”:
$$\color{blue}{x^2+(b+a)x+ab=(x+a)(x+b)}$$

### Example 1:

Factor this trinomial. $$x^2-3x-18=$$

Solution:

Break the expression into groups: $$(x^2+3x)+(−6x−18)$$
Now factor out $$x$$ from $$x^2+3x : x(x+2)$$ , and factor out $$−6$$ from $$−6x+18: −6(x+3)$$
Then: $$=x(x+3)−6(x+3)$$, now factor out like term: $$x+3$$
Then: $$(x+3)(x−6)$$

### Example 2:

Factor this trinomial. $$x^2+x-20=$$

Solution:

Break the expression into groups: $$(x^2-4x)+(5x-20)$$
Now factor out $$x$$ from $$x^2-4x : x(x+3)$$ , and factor out $$5$$ from $$5x-20: 5(x-4)$$
Then: $$=x(x-4)+5(x-4)$$, now factor out like term: $$x-4$$
Then: $$(x+5)(x-4)$$

### Example 3:

Factor this trinomial. $$x^2-2x-8=$$

Solution:

Break the expression into groups: $$(x^2+2x)+(-4x-8)$$
Now factor out x from $$x^2+2x : x(x+2)$$ and factor out $$-4$$ from $$-4x-8: -4(x+2)$$
Then: $$=x(x+2)-4(x+2)$$ , now factor out like term: $$x+2$$
Then: $$(x+2)(x-4)$$

### Example 4:

Factor this trinomial. $$x^2- 6x+8=$$

Solution:

Break the expression into groups: $$(x^2-2x)+(-4x+8)$$
Now factor out x from $$x^2-2x : x(x-2)$$ , and factor out $$-4$$ from $$-4x+8: -4(x-2)$$
Then: $$=x(x-2)-4(x-2)$$, now factor out like term: $$x-2$$
Then: $$(x-2)(x-4)$$

## Exercises

### Factor each Trinomial

1. $$\color{blue}{x^2 – 7x + 12}$$
2. $$\color{blue}{x^2 + 5x – 14}$$
3. $$\color{blue}{x^2 – 11x – 42}$$
4. $$\color{blue}{6x^2 + x – 12}$$
5. $$\color{blue}{x^2 – 17x + 30}$$
6. $$\color{blue}{x^2 + 8x + 15}$$

1. $$\color{blue}{(x – 3) (x – 4)}$$
2. $$\color{blue}{(x – 2) (x + 7)}$$
3. $$\color{blue}{(x + 3) (x – 14)}$$
4. $$\color{blue}{(2x + 3) (3x – 4)}$$
5. $$\color{blue}{(x – 15) (x – 2)}$$
6. $$\color{blue}{(x + 3) (x + 5)}$$

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