#### Need Help?

Get in touch with us

# Factoring Equations

## Key Concepts

• Factor a quadratic trinomial when a is not equal to 1
• Factor out a Common Factor
• Understand factoring by grouping
• Factor a trinomial using substitution

### Factoring a trinomial when a is not equal to 1

#### Factor out a Common Factor

What is the factored form of 3x3 + 15x2 – 18x?

Before factoring the trinomial into two binomials, look for any common factors that you can factor out.

So, 3x3 + 15x2 – 18x = 3x(x2 + 5x – 6).

Then factor the resulting trinomial, x2 + 5x – 6.

The factored form of x2 + 5x – 6 is (x – 1)(x  + 6), so the factored form of 3x3 + 15x2 – 18x is 3x(x – 1)(x + 6).

### Understand factoring by grouping

1. If ax2 + bx + c is a product of binomials, how are the values of a, b and c related? Consider the product (3x + 4)(2x + 1).

The product is 6x2 + 11x + 4. Notice that ac = (6)(4) or (3)(2)(4)(1), which is the product of all of the coefficients and constants from (3x + 4)(2x + 1).

In the middle terms, the coefficients of the x-terms, 3 and 8, add to form b = 11. They are composed of the pairs of the coefficients and constants from the original product; 3 = (3)(1) and 8 = (4)(2).

If ax2 + bx + c is the product of the binomials, there is a pair of the factors of ac that have a sum of b.

1. How can you factor ax2 + bx + c by grouping?

Consider the trinomial 6x2 + 11x + 4, a = 6 and c = 4, so ac = 24.

Find the factor pair of 24 with a sum of 11

The factored form of 6x2 + 11x + 4 is (3x + 4)(2x + 1).

Check. (3x + 4)(2x + 1) = 6x2 + 3x + 8x + 4 = 6x2 + 11x + 4

1. Factoring a trinomial using substitution method

How can you use substitution to help you factor ax2 + bx + c as the product of two binomials?

Consider the trinomial 3x2 – 2x – 8.

Step 1. Multiply ax2 + bx + c by a to transform x2 into (ax)2.

Step 2. Replace ax with a single variable. Let p = ax.

= p2 – 2p – 24

Step 3. Factor the trinomial.

= (p – 6)(p + 4)

Step 4.

Substitute ax back into the product. Remember p = 3x. Factor out the common factors if there are any.

Step 5.

Since you started by multiplying the trinomial by a, you must divide by a to get a product that is equivalent to original trinomial.

The factored form of 3x2 – 2x – 8 is (x – 2)(3x + 4). In general, you can use substitution to help transform ax2 + bx + c with a not equal to 1 to a simpler case in which a = 1, factor it, and then transform it back to an equivalent factored form.

### Questions

Question 1

Write the factored form of each trinomial.

1. 5x2 – 35x + 50

Take out 5.

5(x2 – 7x + 10)

Find the factors of x2 – 7x + 10

Factors of 10 are -5 and -2.

-5 + (-2) = -7

So, (x – 5)(x – 2)

Ans: 5(x – 5)(x – 2)

2. 6x3 + 30x2 + 24x

Take out 6x.

6x(x2 + 5x + 4)

Find the factors of x2 + 5x + 4

Factors of 4 are 4 and 1.

4 + 1 = 5

So, (x + 4)(x + 1)

Ans: 6x(x + 4)(x + 1)

3. 10x2 + 17x + 3

a = 10, b = 17, c = 3

a × c = 10 × 3 = 30

Factors of 30 are 15 and 2.

And 15 + 2 = 17

10x2 + 15x + 2x + 3

= 5x(2x + 3) + 1(2x + 3)

= (5x + 1)(2x + 3)

4. 2x2 – x – 6

a = 2, b = -1, c = -6

a × c = -12

Factors of -12 are -4 and 3.

And (-4) + 3 = -1

2x2 – 4x + 3x – 6

= 2x(x – 2) + 3(x – 2)

= (2x + 3)(x – 2)

5. 10x2 + 3x – 1

a = 10, b = 3, c = -1

a*c = -10

Factors of -10 are 5 and -2.

And 5 + (-2) = 3

10x2 + 5x – 2x – 1

= 5x(2x + 1) + (-1)(2x + 1)

= (5x – 1)(2x + 1)

Question 2

A photographer is placing photos in a mat for a gallery show. Each mat she uses is x in. wide on each side. The total area of each photo and mat is shown below.

1. Factor the total area to find the possible dimensions of a photo and mat.
1. What are the dimensions of the photos in terms of x?

Solution:

1. Total area of the photo and the mat = 4x^2 + 36x + 80

Let’s factor this trinomial using the substitution method.

(2x) 2 + 18(2x) + 80

Let 2x = p.

The trinomial becomes p2 + 18p + 80.

Factors of 80 are 10 and 8.

And 10 + 8 = 18

(p + 10)(p + 8)

As p = 2x,

4x2 + 36x + 80 = (2x + 10)(2x + 8)

2. The dimensions of the photo and the mat combined are 2x + 10 in. by 2x + 8 in.

To find the dimension of just the photo, subtract 2x from both length and width.

L = (2x + 10) – 2x = 10 in.

W = (2x + 8) – 2x = 8 in.

The dimensions of each photo are 10 in. by 8 in. In terms of x, it is 10x0 in. by 8x0 in. (independent of the value of x).

## Exercise

Factor the following trinomials:

1. 8x2 – 10x – 3
2. 12x2+ 16x + 5
3. 16x3 + 32x2 + 12x
4. 21x2 – 35x – 14
5. 16x2 + 22x – 3
6. 9x2 + 46x + 5
7. –6x2 – 25x – 25
8. 5x2 – 4xy – y2
9. 16x2 + 60x – 100
10. 6x2 + 5x – 6

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   