Trinomials: Factoring, Solving & Examples

Instructor: DaQuita Hester

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

Trinomials can have different leading coefficients and therefore, have to be factored differently. Learn how to factor and solve any trinomial. Then, test your knowledge with a quiz.

Definition

A trinomial is an equation that consists of three terms. For this lesson, we will examine trinomials written in the form ax^2 + bx + c, where a, the leading coefficient, does not equal zero.

To factor a trinomial means to rewrite it as a product of two binomials. This means that we are going to rewrite the trinomial in the form (x + m) (x + n). Your task is to determine the value of m and n.

For factoring, trinomials are divided into two groups: those with a leading coefficient of 1 and those with a leading coefficient not equal to 1. Let's examine both.

Factoring Trinomials with a Leading Coefficient of 1

Use the following steps to factor the trinomial x^2 + 7x + 12.

Step 1: Determine the factor pairs of c that will add to get b. For x^2 + 7x + 12, a = 1, b = 7, and c = 12. So to complete this step, we have to figure out which factor pairs of 12 will add together to equal 7. Let's list them. The factor pairs are 1 & 12, 2 & 6, and 3 & 4. The third pair is what we need, because the sum of these two numbers is 7, which is our b.

Step 2: In separate parentheses, add each number to x. Here, we are simply going to take our factor pair and add each one to x. When we do, we get (x + 3) and (x + 4). Therefore, x^2 + 7x +12 factored is (x + 3)(x + 4).

To ensure that we've factored correctly, let's multiply (x + 3) by (x + 4) to see if we get our original trinomial. When we distribute, we see that (x)(x) = x^2, (x)(4) = 4x, (3)(x) = 3x, and (3)(4) = 12. By combining our like terms, we get x^2 + 7x +12, which was our original trinomial. Therefore, we can conclude that our factoring was done correctly.

Factoring Trinomials with a Leading Coefficient Not Equal to 1

To factor trinomials with a not equal to 1, the process will be a little different. Let's walk through the steps below and use them to factor 2x^2 - 5x - 3.

Step 1: Multiply a and c together. For this trinomial, a = 2, b = - 5, and c = -3. When we multiply a and c, we get (2)(-3) = -6.

Step 2: Identify the factor pairs of this product that will add together to equal b. To complete this step, we must list the factor pairs of -6. They are -1 & 6, 1 & -6, 2 & -3, and -2 & 3. The pair that will add together to get b, which is - 5, is 1 and -6.

Step 3: Re-write the original equation, but replace b with the correct factor pair. Since -5 = 1 - 6, we will replace -5x with 1x - 6x. This gives us 2x^2 + x - 6x - 3.

Step 4: Group the equation into two parentheses, each with two terms. Then, factor each one. Our two parentheses will be (2x^2 + x) and (-6x - 3). Let's factor them. From the first set of parentheses, we can factor out an x to get x*(2x + 1). In the second parenthesis, let's factor out -3. This leaves us with -3*(2x + 1). Now we can see that 2x^2 - 5x - 3 = (2x^2 + x)+(-6x - 3) = x*(2x + 1) + -3*(2x + 1). Notice that the equations left inside of both parentheses are the same. If we are factoring correctly, this should happen every time.

Step 5: Place the factored terms into a separate parenthesis. The terms that we factored out of the parenthesis were x and -3. By placing them into their own parenthesis, we end up with (x - 3) and (2x + 1) as the factors of 2x^2 - 5x - 3.

Let's see if we did this correctly and multiply (x - 3) by (2x + 1). From distributing, we see that (x)(2x) = 2x^2, (x)(1) = x, (-3)(2x) = -6x, and (-3)(1) = -3. After combining like terms, we see that the product is 2x^2 - 5x - 3, which is what we started with. We can now be sure that we have factored correctly.

Solving Trinomials

Now that we are able to factor trinomials, let's practice solving them. To do so, we want ax^2 + bx + c to equal zero.

Take a look at the equation 3x^2 - 5x - 5 = -3.

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