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Factor Quadratics: x^2 + bx + c

Introduction

The most basic kind of factoring involves taking an expression and finding the greatest common factor of each of the terms in the expression.  In the example to the right, each expression has a common factor of 3x, and this term can be “factored” out of the original expression.

 

The three terms in the expression y2 + 8y + 12 do not have any common factors.  This expression can be factored using other methods and the result is (y + 6)(y + 2).

The first step in understanding factoring is understanding how to multiply the factors (y + 6) and (y + 2) to get the original expression above.

 

Factoring_a_Trinomial_vis_2

The lesson shows strategies of how to factor trinomials like y2 + 8y + 12.  Before moving on, take a minute and look carefully at the multiplication of the (y + 6) and (y + 2) above.  Can you guess how the 8y and the 12 (in y2 + 8y + 12) relate to the (y + 6) and (y + 2)?  

Lesson

In the introduction, (y + 6) and (y + 2) were multiplied to get an answer of y2 + 8y + 12.  You may notice that:

  • 6 + 2 = 8 (our 8y)
  • 6 ∙ 2 = 12 (our 12)

These two facts can be used to factor many trinomials.

 

Example 1: Factor the trinomials

Factoring_a_Trinomial_vis_3a

Factoring_a_Trinomial_vis_4a

Factoring_a_Trinomial_vis_5a

Factoring_a_Trinomial_vis_6a

*Scroll over the problem above to reveal the answer.

 

Each trinomial in example 1 is in the form ax2 + bx + c, where a = 1 (in other words, x2 + bx + c).  When factoring trinomials of this type, it is just a matter of finding two numbers that add up to b and multiply to equal c. 

 

In example 1, each number was a positive number.  Now that you have had a little practice factoring trinomials, see if you can do a problem that involves negatives:

Factoring_a_Trinomial_vis_7aFactoring_a_Trinomial_vis_7b1

Working with negatives simply means more possibilities when finding factors.  In example 2, some of the numbers are negative.  See if you can figure out the answer, then scroll over it to see if you were correct.

 

Example 2: Factor the trinomials

 

Factoring_a_Trinomial_vis_8a

Factoring_a_Trinomial_vis_9a

Factoring_a_Trinomial_vis_10a

 Factoring_a_Trinomial_vis_11a

*Scroll over the problem above to reveal the answer.

 

Special Cases in Factoring

Consider the expressions x2 + 1 and x2 – 1.  Only one of these is factorable.  Can you guess which one it is?

 

Factoring_a_Trinomial_vis_12You may think that x2 + 1 would factor to (x + 1)(x + 1), but the math just doesn’t work out.  The expression x2 + 1 is actually the same as x2 + 0x + 1.  To factor it, there would need to be two numbers that multiplied to 1 and added to zero.  Since there are no such numbers, it is nonfactorable.

 

The expression x2 – 1 can also be written as x2 + 0x – 1.  This can be factored and the answer must contain two numbers that multiply to -1 and add to zero.  The only two integers that multiply to -1 are 1 and -1.  These two numbers add up to 0, so they work for factoring x2 – 1.  This is called the difference of two squares.

 

Factoring_a_Trinomial_vis_13

 

The chart above shows how the factors (x + 1) and (x – 1) can be combined to make these special cases.  The same works for (x + a) and (x – a), where a is any positive whole number.

Example 3: Factor each 2nd degree polynomial.  Write “not factorable” if it cannot be factored.

Factoring_a_Trinomial_vis_14a

Factoring_a_Trinomial_vis_15a

Factoring_a_Trinomial_vis_16a

Factoring_a_Trinomial_vis_17a

*Scroll over the problem above to reveal the answer.

Try It

Factor each polynomial.  Write “not factorable” if it cannot be factored.

 

1)  x2 + 10x + 16

2)  x2 – 25

3)  y2 – y – 20

4)  y2 – 100

5)  z2 + 11z + 10

6)  z2 + 36

7)  a2 – 18x + 80

8)  a2 + 11a – 60

*Hint:  It is easy to peek at the answers below, so it is recommended that you try all problems on a separate piece of paper, then check answers all at once.

 

 

 

Scroll down for answers:

 

 

 

 

 

 

 

 

 

 

Answers:

1)  x2 + 10x + 16  =  (x + 8)(x + 2)

2)  x2 – 25  =  (x + 5)(x – 5)

3)  y2 – y – 20  =  (y + 4)(y – 5)

4)  y2 – 100  =  (y + 10)(y – 10)

5)  z2 + 11z + 10  =  (z + 10)(z + 1)

6)  z2 + 36  »  not factorable

7)  a2 – 18x + 80  =  (a – 10)(z – 8)

 

8)  a2 + 11a – 60  =  (a + 15)(a – 4)


 

 

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