6.2: Factoring Trinomials of the Form x²+bx+c
- Last updated
- Sep 2, 2024
- Save as PDF
( \newcommand{\kernel}{\mathrm{null}\,}\)
- Factor trinomials of the form
- Factor trinomials using the AC method.
Factoring Trinomials of the Form
Some trinomials of the form
We can verify this factorization by multiplying:
Factoring trinomials requires that we work the distributive process in reverse. Notice that the product of the first terms of each binomial is equal to the first term of the trinomial.
The middle term of the trinomial,
And the product of the last terms of each binomial is equal to the last term of the trinomial.
This can be visually interpreted as follows:
.png?revision=1)
If a trinomial of this type factors, then these relationships will be true:
This gives us
In short, if the leading coefficient of a factorable trinomial is one, then the factors of the last term must add up to the coefficient of the middle term. This observation is the key to factoring trinomials using the technique known astrial and error (or guess and check). The steps are outlined in the following example.
Factor:
Solution:
Note that the polynomial to be factored has three terms; it is a trinomial with a leading coefficient of
Step 1: Write two sets of blank parentheses. If a trinomial of this form factors, then it will factor into two linear binomial factors.
Step 2: Write the factors of the first term in the first space of each set of parentheses. In this case, factor
Step 3: Determine the factors of the last term whose sum equals the coefficient of the middle term. To do this, list all of the factorizations of
Choose
Step 4: Write in the last term of each binomial using the factors determined in the previous step.
Step 5: Check by multiplying the two binomials.
Answer:
Since multiplication is commutative, the order of the factors does not matter.
If the last term of the trinomial is positive, then either both of the constant factors must be negative or both must be positive. Therefore, when looking at the list of factorizations of the last term, we are searching for sums that are equal to the coefficient of the middle term.
Factor:
Solution:
First, factor
Next, determine which factors of
In this case, choose
Check.
Answer:
If the last term of the trinomial is negative, then one of its factors must be negative. In this case, search the list of factorizations of the last term for differences that equal the coefficient of the middle term.
Factor:
Solution:
Begin by factoring the first term
The factors of
Here choose the factors
Multiply to check.
Answer:
Often our first guess will not produce a correct factorization. This process may require repeated trials. For this reason, the check is very important and is not optional.
Factor:
Solution:
The first term of this trinomial
\(x^{2}+5x-6=(x\quad ?)(x\quad ?)
Consider the factors of
Suppose we choose the factors
When we multiply to check, we find the error.
In this case, the middle term is correct but the last term is not. Since the last term in the original expression is negative, we need to choose factors that are opposite in sign. Therefore, we must try again. This time we choose the factors
Now the check shows that this factorization is correct.
Answer:
If we choose the factors wisely, then we can reduce much of the guesswork in this process. However, if a guess is not correct, do not get discouraged; just try a different set of factors.
Factor:
Solution:
Here there are no factors of
Answer:
Prime
Factor:
- Answer
-
The techniques described can also be used to factor trinomials with more than one variable.
Factor:
Solution:
The first term
Next, look for factors of the coefficient of the last term,
Therefore, the coefficient of the last term can be factored
Multiply to check.
Visually, we have the following:
.png?revision=1)
Answer:
Factor:
- Answer
-
Factoring Using the AC Method
An alternate technique for factoring trinomials, called the AC method, makes use of the grouping method for factoring four-term polynomials. If a trinomial in the form
Factor using the AC method:
Solution:
In this example
Step 1: Determine the product
Step 2: Find factors of
We can see that the sum of the factors
Step 3: Use the factors as coefficients for the terms that replace the middle term. Here
Step 4: Factor the equivalent expression by grouping.
Answer:
Notice that the AC method is consistent with the trial and error method. Both methods require that
Factor:
Solution:
Here
Therefore,
Answer:
At this point, it is recommended that the reader stop and factor as many trinomials of the form
Key Takeaways
- Factor a trinomial by systematically guessing what factors give two binomials whose product is the original trinomial.
- If a trinomial of the form
- Not all trinomials can be factored as the product of binomials with integer coefficients. In this case, we call it a prime trinomial.
- Factoring is one of the more important skills required in algebra. For this reason, you should practice working as many problems as it takes to become proficient.
Are the following factored correctly? Check by multiplying.
- Answer
-
1. No
3. Yes
5. Yes
7. No
9. Yes
Factor.
- The area of a square is given by the function
is measured in meters. Rewrite this function in factored form. - The area of a square is given by the function
is measured in meters. Rewrite this function in factored form.
- Answer
-
1.
3.
5.
7.
9.
11.
13. Prime
15.
17.
19. Prime
21.
23.
25.
27.
29. Prime
31.
33.
35.
37.
39.
41.
43.
45.
45.
Factor using the AC method.
- Answer
-
1.
3.
5.
7.
9. Prime
11.
- Create your own trinomial of the form
- Write out your own list of steps for factoring a trinomial of the form
- Create a trinomial that does not factor and share it along with an explanation of why it does not factor.
- Answer
-
1. Answers may vary
3. Answers may vary
