Factoring x² + bx + c

Key Concepts

• Understand factoring a trinomial
• Factor x² + bx + c, when b < 0 and c < 0
• Factor x² + bx + c, when c <0
• Factor a trinomial with two variables
• Apply factoring trinomials
  

Understand factoring a trinomial 

Concept 

1. How does factoring a trinomial relate to multiplying binomials?  

Consider the binomial product (x + 2) (x + 3) and the trinomial x² + 5x + 6.  

When factoring a trinomial, you work backward trying to find the related binomial factors whose product equals the trinomial.  

You can factor the trinomial of the form x² + bx + c as (x + p) (x + q) if pq = c and p + q = b.  

2. What is the factored form of x² + 5x + 6? 

 Identify a factor pair of 6 that has a sum of 5.   

If you factor using algebra tiles, the correct factor pair will form a rectangle.  

The factored form of x² + 5x + 6 is (x + 2) (x + 3).  

The first term of each binomial is x, since x * x = x² 

Check (x + 2) (x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 

Factor x² + bx + c, when b < 0 and c > 0   

What is the factored form of x² – 11x + 18?  

Identify a factor pair of 18 that has a sum of -11.   

The factored form of x² – 11x + 18 is (x – 2)(x – 9). 

Check (x – 2) (x – 9) = x² – 9x – 2x + 18 = x² – 11x + 18  

Factor x² + bx + c, when c < 0   

What is the factored form of x² + 5x – 6?  

Identify a factor pair of -6 that has a sum of 5.   

The factored form of x² + 5x – 6 is (x – 1) (x + 6).                                                      

Factor a Trinomial with Two Variables  

Concept 

1. How does multiplying binomials in two variables relate to factoring trinomials? 

Consider the following binomial products.  

(x – 2y) (x + 4y) = x² + 6xy + 8y²  

(x – 3y) (x + 5y) = x² + 2xy – 15y²    

(x – 7y) (x – 9y) = x² – 16xy – 63y²   

Each trinomial has the form of x² + bxy + cy². Trinomials of this form are factorable when there is a factor pair of c  that has a sum of b.  

2. What is the factored form of x² + 10xy + 24y²? 

Identify a factor pair of 24 that has a sum of 10.   

The factored form of x² + 10xy + 24y² is (x + 4y) (x + 6y).  

Check (x + 4y) (x + 6y) = x² + 6xy + 4xy + 24y² = x² + 10xy + 24y²                                                     

Apply Factoring Trinomials 

1. Example 

Benjamin is designing a new house. The bedroom closet will have one wall that contains a closet system using three different-sized storage units. The number and amount of wall space needed for each of the three types of storage units is shown. What are the dimensions of the largest amount of wall space that will be needed? 

Solution:  

Formulate:  

The largest possible closet system will use all of the units. Write an expression that represents the wall area of the closet in terms of the storage units. 

x² + 12x + 35  

Compute:  

Because the area of the rectangle is the product of the length and the width, factor the expression to find binomials that represent the length and the width of the closet wall.  

x² + 12x + 35 = (x + 5) (x + 7)  

Interpret:  

The dimensions of the largest amount of wall space that will be needed are (x + 7) ft. by (x + 5) ft.  

Questions  

Question 1 

Write the factored form of each trinomial.  

1. x² + 13x + 36  

Factors of 36 are 9 and 4. 

And the sum of 9 and 4 is 13.  

(x + 9) (x + 4)  

2. x² – 8x + 15  

Factors of 15 are -3 and -5.  

And the sum of (-3) and (-5) is (-8).  

(x – 3) (x – 5)  

3. x² – 5x – 14 

Factors of -14 are 2 and -7.  

And the sum of 2 and (-7) is -5.  

(x + 2) (x – 7)  

4. x² + 6x – 16   

Factors of -16 are 8 and -2.  

And the sum of 8 and -2 is 6.  

(x + 8) (x – 2)  

5. x² + 12xy + 32y² 

Factors of 32 are 8 and 4.  

And the sum of 8 and 4 is 12.  

(x + 8y) (x + 4y)  

Question 2 

In the last example mentioned on the page – 4, what would be the dimensions of the largest wall you would need, if you used 11 of the 1 ft-by-1 ft units while keeping the other units the same? 

Solution:  

Formulate: Write an expression that represents the wall area of the closet in terms of the storage units. 

x² + 12x + 11 

Compute: Because the area of the rectangle is the product of the length and the width, factor the expression to find binomials that represent the length and the width of the closet wall.  

Factors of 11 are 11 and 1.  

And the sum of 11 and 1 is 12.  

So, x² + 12x + 11 = (x + 11) (x + 1)  

Interpret: The dimensions of the largest amount of wall space that will be needed are (x + 11) ft. by (x + 1) ft.  

Key Concepts Covered  

To factor a trinomial of the form x² + bx + c, find a factor pair of c that has a sum of b. Then use the factors you found to write the binomials that have a product equal to the trinomial.  

Exercise

Factor the following trinomials:

1. x²+9x+20
2. x²+8x+12
3. x²-7x-18
4. x²-6x+9
5. x²+ 2xy + y²
6. x²-18x+81
7. 3x²+ 39x-90
8. x²– 6xy + 9y²
9. x²-x-6
10. x²– 13x – 30