Factoring Trinomials (2024)

Factoring truly can be one of the most challenging skills that students learn in secondary mathematics. To make matters worse, it is used very often in more complex mathematics. This is why it is so important that teachers explore how to teach factoring in ways that support student learning. There are so many different factoring methods out there that it is important to think about the best one for you. Whether you are a teacher or a student, finding a strategy that helps you feel comfortable is important. [1]

Contents

1. Background
   1. Trinomial
   2. Perfect Square Trinomial
   3. Quadratic Trinomial
   4. Factoring Trinomials
2. The Factoring Decision Tree (Math By The Pixel Way)
   1. Step 1: Check for a Greatest Common Factor
   2. Step 2: Is the Quadratic Expression a Trinomial?
      1. Differences of Squares?
         1. Difference-of-Squares Formula
      2. Grouping?
   3. Step 3: Check the Leading Coefficient
      1. When the Leading Coefficient Is Equal to One
      2. When the Leading Coefficient Is Not Equal to One
   4. Step 4: Checking Your Work
3. Other Factoring Methods
   1. Box Method
   2. Magic Squares
   3. Perfect Square
   4. Diamond Method
   5. Quadratic Equation
4. Factor Trees
5. FOIL
6. References
7. Additional Reading
8. Videos

Background

In mathematics, monomials, binomials, trinomials and polynomials are all algebraic expressions. The expressions that are represented using unknown variables, constants and coefficients, are calledalgebraic expressions. A variable can take any value, it is not fixed but a constant is a fixed value.

Trinomial

A trinomial is an expression which is composed of exactly three terms.

The parts of algebraic expressions that are separated by addition or subtraction, are called terms. The number of terms decides the type of expression, whether it is a monomial, binomial, trinomial or polynomial. These terms are made of a product of variables and coefficients that are added together to form expressions. [2]

Perfect Square Trinomial

A perfect square trinomial is a mathematical expression formed by squaring abinomial. It follows the pattern ax²+ bx + c, where (a), (b), and (c) arereal numbers, and (a) is not equal to zero. It also meets the condition b²= 4ac. Perfect Square Trinomials Formulaare,

• (ax)²+ 2abx + b²= (ax + b)²
• (ax)²– 2abx + b²= (ax – b)²

To recognize a perfect square trinomial, follow these steps:

Check Form:Look at the expression, and if it’s in the form (ax²+ bx + c), it could be a perfect square trinomial.

Verify Condition:Confirm if the condition b²= 4ac is met. Here, (b) is the coefficient of the linear term, and (a) and (c) are the coefficients of the squared and constant terms, respectively.

Compare with Formula:See if the expression matches the structure of (ax + b)²or (ax – b)². If it does, then it’s a perfect square trinomial. [3]

Quadratic Trinomial

Aquadratictrinomial is a specific kind of mathematical expression containing both variables and constants. It appears in the form (ax²+ bx + c), where (x) is the variable, and (a), (b), and (c) are real numbers that are not zero. Here, (a) is called the leading coefficient, (b) is the linear coefficient, and (c) is the additive constant.

There’s a key aspect related to quadratic trinomials, called the discriminant (D), expressed as (D = b²– 4ac). The discriminant helps categorize different cases of quadratic trinomials. By evaluating (D), you can understand more about the nature of the quadratic expression.

If a quadratic trinomial, which involves just one variable, equals zero, it transforms into what’s known as a quadratic equation, represented as (ax²+ bx + c = 0). In simpler terms, when a quadratic trinomial takes this form, it becomes a quadratic equation. [3]

Factoring Trinomials

Factoring trinomialsmeans transforming a mathematical expression from having three terms to having two terms. A trinomial is a polynomial with three terms, generally represented as ax²+bx+c, where a and b are coefficients, and c is a constant.

To factor a trinomial, twointegers, often denoted as r and s, are selected such that their sum equals b, and their product equals ac. Trinomial is then rewritten as ax²+ rx + sx + c. Using distributive property, the polynomial is then factored. Now the trinomial is written as (x + r)(x + s). [3]

The Factoring Decision Tree (Math By The Pixel Way)

Giving students something visual that helps them see the decisions they have available to them is very helpful and makes the whole process much less stressful for them. [1]

Step 1: Check for a Greatest Common Factor

Remember if the leading coefficient is negative, so is the GCF.

−10y⁴ − 55y³ − 60y²

-5y² (2y² +11y + 12)

Step 2: Is the Quadratic Expression a Trinomial?

A trinomial is an expression which is composed of exactly three terms.

Differences of Squares?

When you learn tofactor quadratics, there are three other formulas that they usually introduce at the same time. The first is the “difference of squares” formula.

Remember from yourtranslationskills that a “difference” means a “subtraction”. So a difference of squares is something that looks likex²− 4. That’s because4 = 2², so we really havex²− 2², which is a difference of squares. [5]

To factor this, I’ll start by writing my parentheses, in the same way as usual for factoring:

x²− 4 = (x )(x )

For this quadratic factorization, I need factors of−4that add up to zero, so I’ll use−2and+2:

x²− 4 = (x− 2)(x+ 2)

(ReviewFactoring Quadratics, if the steps in this example didn’t make sense to you.)

Difference-of-Squares Formula

For a difference of squaresa²−b², the factorization is: (a−b)(a+b)

Grouping?

Another way to factor trinomials of the form𝑎𝑥²+𝑏𝑥+𝑐is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods used in factoring trinomials with a leading coefficient of one. This method is very structured (that is step-by-step), and it always works!

Factor using the‘ac’method:6x²+7x+2. [4]

Step 3: Check the Leading Coefficient

The leading coefficient is the number in front of the x termthat has an exponent of two(if you are working with a quadratic expression).

When the Leading Coefficient Is Equal to One

If the leading coefficient is equal to one, we are working with a simple case of trinomial factoring. We can start this process with decomposition, or the “splitting the middle term” process. [1]

When the Leading Coefficient Is Not Equal to One

If the leading coefficient is anything other than one, I direct my students to the right side of the decision tree above, i.e., Differences of Squares and Grouping. The same process as above can be applied to tackle a problem like this. I have my own crazy trick for factoring these sorts of trinomials that you sort of have to see to believe. You can check out avideo of my trinomial factoring process. [1]

Step 4: Checking Your Work

After successfully factoring a quadratic polynomial, I always make sure to check if my answer makes sense. The only way to do this is to apply the distributive property for binomials (sometimes known as FOIL). This process allows us to take a product of binomials and write them in standard form. If you see that your standard form expression is the same as the original trinomial, you know that your factored expression is correct! [1]

Other Factoring Methods

Box Method

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Magic Squares

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Perfect Square

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Diamond Method

ASAS

Quadratic Equation

See the Quadratic Equation webpage.

Factor Trees

Factor treesare a way of expressing the factors of a number, specifically the prime factorization of a number. [6]

Each branch in the tree is split into factors. Once the factor at the end of the branch is aprime number(only one pair of factors, the number itself and11), the branch stops and youcircle the number.

Factor trees include whole numbers. However, since11is not a prime numberit will not appear in any factor tree.

To complete factor trees, it is helpful to remember the prime numbers from1−20:
2,3,5,7,11,13,17,192,3,5,7,11,13,17,19, …

FOIL

FOIL is a mnemonic for the standard method of multiplying two binomials, hence the method may be referred to as the FOIL method.

The word FOIL is an acronym for the four terms of the product: [7]

• First (“first” terms of each binomial are multiplied together)
• Outer (“outside” terms are multiplied—that is, the first term of the first binomial and the second term of the second)
• Inner (“inside” terms are multiplied—second term of the first binomial and first term of the second)
• Last (“last” terms of each binomial are multiplied)

The general form is:

References

[1] Jordan. 2024. “How to Teach Factoring Trinomials (The Best Method!).”Math By The Pixel. May 23. https://mathbythepixel.com/how-to-teach-factoring/.

[2] “Monomials, Binomials, Trinomials and Polynomials.”2021. BYJUS. BYJU’S. July 13. https://byjus.com/maths/monomials-binomials-trinomials-and-polynomials/.

[3] “Trinomials: Definition, Formula, Types, and Examples.”2024. GeeksforGeeks. GeeksforGeeks. May 30. https://www.geeksforgeeks.org/trinomials/.

[4] “6.3: Factor Trinomials.”2024. Mathematics LibreTexts. Libretexts. February 19. https://math.libretexts.org/Workbench/Intermediate_Algebra_2e_(OpenStax)/06%3A_Factoring/6.03%3A_Factor_Trinomials.

[5] Stapel, Elizabeth. 2024. “Special Factoring: Differences of Squares.”Purplemath. Accessed June 19. https://www.purplemath.com/modules/specfact.htm.

[6] “Factor Trees – Math Steps, Examples & Questions.” 2024.Third Space Learning. February 24. https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/factor-tree/.

[7] Admin. 2022. “Foil Formula Algebra: What Is the Foil Method in Math.”BYJUS. BYJU’S. https://byjus.com/foil-formula/.

Additional Reading

“Difference of Squares | Factoring Quadratics (Article).” 2024.Khan Academy. Khan Academy. Accessed June 19. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-difference-squares/a/factoring-quadratics-difference-of-squares.

“Polynomials (Definition, Types and Examples).”2023. BYJUS. BYJU’S. October 3. https://byjus.com/maths/polynomial/.

“Trinomials – Definition, Types, Formulas, Examples.”2024. examples.com. Examples.com. May 22. https://www.examples.com/maths/trinomials.html.

“Trinomials – Formula, Examples, Types.” 2024.CUEMATH. Accessed June 19. https://www.cuemath.com/algebra/trinomial/.

“Trinomials: Formula, Types, Identities, Factor with Examples.” 2024.Testbook. Accessed June 19. https://testbook.com/maths/trinomial.

Videos

This video is a walkthrough of factoring difference of squares. How does it work? The answer is simpler than you think! Factoring a difference of two squares can be done by imagining that the middle term of the expression is 0x. From here, regular trinomial factoring can be applied! It is as simple as that!

A walkthrough of a quick trinomial factoring shortcut that can be used to factor complex trinomials. Trinomial factoring is one of the concepts students tend to struggle with the most. This insane trinomial factoring shortcut will blow your mind! You need to see it to believe it! It works every time! In this video I will show you how to factor trinomials the easy way by explaining a trinomial factoring shortcut I have been using throughout my entire mathematics career. This method is quick, accurate, and effective, and saves you the work of using decomposition or magic squares. This is particularly effective when you want to know how to factor trinomials when a isn’t 1 (or how to factor trinomials with coefficients).

Splitting the Middle Term

The featured image on this page is from the Math By The Pixel website.