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Guias de estudo > Intermediate Algebra

Read: Factor a Trinomial with Leading Coefficient = 1

Learning Objectives

  • Identify a trinomial
  • Identify the leading coefficient of a trinomial
  • Use a method to factor a trinomial with a leading coefficient of 11
Trinomials are polynomials with three terms. We are going to show you a method for factoring a trinomial whose leading coefficient is 11.  Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that trinomials can be factored. The trinomial x2+5x+6{x}^{2}+5x+6 has a GCF of 11, but it can be written as the product of the factors (x+2)\left(x+2\right) and (x+3)\left(x+3\right). Recall how to use the distributive property to multiply two binomials:

(x+2)(x+3)=x2+3x+2x+6=x2+5x+6\left(x+2\right)\left(x+3\right) = x^2+3x+2x+6=x^2+5x+6

We can reverse the distributive property and return x2+5x+6 to (x+2)(x+3)x^2+5x+6\text{ to }\left(x+2\right)\left(x+3\right)  by finding two numbers with a product of 66 and a sum of 55.

Factoring a Trinomial with Leading Coefficient 1

In general, for a trinomial of the form x2+bx+c{x}^{2}+bx+c you can factor a trinomial with leading coefficient 11 by finding two numbers, pp and qq whose product is c, and whose sum is b.
Let's put this idea to practice with the following example.

Example

Factor x2+2x−15{x}^{2}+2x - 15.

Answer: We have a trinomial with leading coefficient 1,b=21,b=2, and c=−15c=-15. We need to find two numbers with a product of −15-15 and a sum of 22. In the table, we list factors until we find a pair with the desired sum.

Factors of −15-15 Sum of Factors
1,−151,-15 −14-14
−1,15-1,15 1414
3,−53,-5 −2-2
−3,5-3,5 22
Now that we have identified pp and qq as −3-3 and 55, write the factored form as (x−3)(x+5)\left(x - 3\right)\left(x+5\right).

In the following video we present two more examples of factoring a trinomial with a leading coefficient of 1. https://youtu.be/-SVBVVYVNTM   To summarize our process consider these steps:

How To: Given a trinomial in the form x2+bx+c{x}^{2}+bx+c, factor it.

  1. List factors of cc.
  2. Find pp and qq, a pair of factors of cc with a sum of bb.
  3. Write the factored expression (x+p)(x+q)\left(x+p\right)\left(x+q\right).
We will now show an example where the trinomial has a negative c term. Pay attention to the signs of the numbers that are considered for p and q.
In our next example, we show that when c is negative, either p or q will be negative.

Example

Factor x2+x–12x^{2}+x–12.

Answer: Consider all the combinations of numbers whose product is −12-12, and list their sum.

Factors whose product is −12−12 Sum of the factors
1⋅−12=−121\cdot−12=−12 1+−12=−111+−12=−11
2⋅−6=−122\cdot−6=−12 2+−6=−42+−6=−4
3⋅−4=−123\cdot−4=−12 3+−4=−13+−4=−1
4⋅−3=−124\cdot−3=−12 4+−3=14+−3=1
6⋅−2=−126\cdot−2=−12 6+−2=46+−2=4
12⋅−1=−1212\cdot−1=−12 12+−1=1112+−1=11
Choose the values whose sum is +1+1:  r=4r=4 and s=−3s=−3, and place them into a product of binomials.  

(x+4)(x−3)\left(x+4\right)\left(x-3\right)

Answer

(x+4)(x−3)\left(x+4\right)\left(x-3\right)

 

Think About It

Which property of multiplication can be used to describe why (x+4)(x−3)=(x−3)(x+4)\left(x+4\right)\left(x-3\right) =\left(x-3\right)\left(x+4\right). Use the textbox below to write down your ideas before you look at the answer. [practice-area rows="2"][/practice-area]

Answer: The commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

a⋅b=b⋅aa\cdot b=b\cdot a

In our last example we will show how to factor a trinomial whose b term is negative.

Example

Factor x2−7x+6{x}^{2}-7x+6.

Answer: List the factors of 66. Note that the b term is negative - so we will need to consider negative numbers in our list.

Factors of 66 Sum of Factors
1,61,6 77
2,32, 3 55
−1,−6-1, -6 −7-7
−2,−3-2, -3 −5-5
Choose the pair that sum to −7-7, which is −1,−6-1, -6 Write the pair as constant terms in a product of binomials. (x−1)(x−6)\left(x-1\right)\left(x-6\right)

 Analysis of the solution

In the last example, the b term was negative and the c term was positive. This will always mean that if it can be factored, p and q will both be negative.

Think About It

Can every trinomial be factored as a product of binomials? Mathematicians often use a counter example to prove or disprove a question. A counter example means you provide an example where a proposed rule or definition is not true. Can you create a trinomial with leading coefficient 11 that cannot be factored as a product of binomials? Use the textbox below to write your ideas. [practice-area rows="2"][/practice-area]

Answer: Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime. A counter-example would be: x2+3x+7x^2+3x+7

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