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Study Guides > Intermediate Algebra

Factor a Trinomial with Leading Coefficient = 1

Learning Outcomes

  • Factor a trinomial with leading coefficient =1= 1
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 us put this idea to practice with the following example.

Example

Factor x2+2x15{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 (x3)(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 the following 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+x12x^{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
112=121\cdot−12=−12 1+12=111+−12=−11
26=122\cdot−6=−12 2+6=42+−6=−4
34=123\cdot−4=−12 3+4=13+−4=−1
43=124\cdot−3=−12 4+3=14+−3=1
62=126\cdot−2=−12 6+2=46+−2=4
121=1212\cdot−1=−12 12+1=1112+−1=11
Choose the values whose sum is +1+1:  p=4p=4 and q=3q=−3, and place them into a product of binomials.  

(x+4)(x3)\left(x+4\right)\left(x-3\right)

Think About It

Which property of multiplication can be used to describe why (x+4)(x3)=(x3)(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 factors may be multiplied in any order without affecting the product.

ab=baa\cdot b=b\cdot a

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

Example

Factor x27x+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. (x1)(x6)\left(x-1\right)\left(x-6\right)

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 counterexample to prove or disprove a question. A counterexample 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 counterexample would be: x2+3x+7x^2+3x+7

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