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.  

2. 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   

3. 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.  

2. 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).  

Key Concepts Covered  

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