Factoring Trinomials

Factoring trinomials means writing an expression as the product of two or more binomials and is written as (x + m) (x + n). A binomial is a two-term polynomial whereas a trinomial is a three-term polynomial. Factoring trinomials is done by splitting the algebraic expressions into a binomial that can be multiplied back to a trinomial. Let us know more about factoring trinomials, different methods and solve a few examples to understand the concept better.

Factoring trinomials is converting an algebraic expression from a trinomial expression to a binomial expression. A trinomial is a polynomial with three terms with the general expression as ax² + bx + c, where a and b are coefficients and c is a constant. There are three simple steps to remember while factoring trinomials:

• Identify the values of b (middle term) and c (last term).
• Find two numbers that add to b and multiply to c.
• Use these numbers to factor the expression to obtain the factored terms.

Two integers such as r and s are considered to factor a trinomial, whose sum is b and whose product is ac. We can rewrite the trinomial as ax² + rx + sx + c and then use grouping and the distributive property to factor the polynomial. After the trinomial is undergone the process of factoring, the expression becomes a binomial in the form (x + r) (x + s). Here is an image to understand this better.

For factoring a trinomial there are points or rules to remember. These rules are based on mathematical signs such as (+) and (-) that play an important role while factoring trinomials and make it simple in factoring trinomials. The rules are as follows:

• If all terms of the trinomial are positive, then all terms of the binomials will be positive.
• If the last term of the trinomial is negative but the middle term and the first term are positive, then one term of the binomial will be negative and the other will be positive. (The greater factor will be positive and the smaller will be negative).
• If the middle term and the last term of the trinomial are negative and the first term is positive, then the sign for one binomial will be positive and the other will be negative. (The greater factor will be negative and the smaller will be positive).
• If the last term and the first term of the trinomial are positive but the middle term is negative, then both signs of the binomials will be negative.
• Look for common factors for the trinomial ax² + bx + c, where a is 1. First factor the common factor then factor the rest of the expression.
• If ax² is negative in a trinomial, you can factor −1 out of the whole trinomial first.

Factoring a trinomial means expanding an equation into the product of two or more binomials. It is written as (x + m) (x + n). A trinomial can be factorized in many ways. Let's discuss each case.

Quadratic Trinomial in One Variable

The general form of quadratic trinomial formula in one variable is ax² + bx + c, where a, b, c are constant terms and neither a, b, or c is zero. For the value of a, b, c, if b² - 4ac > 0, then we can always factorize a quadratic trinomial. It means that ax² + bx + c = a(x + h)(x + k), where h and k are real numbers. Now let's learn how to factorize a quadratic trinomial with an example.

Example: Factorize: 3x² - 4x - 4

Solution:

Step 1:- First multiply the coefficient of x² and the constant term.

3 × -4 = -12

Step 2:- Break the middle term -4x such that on multiplying the resulting numbers, we get the result -12 (obtained from the first step).

-4x = -6x + 2x

-6 × 2 = -12

Step 3:- Rewrite the main equation by applying the change in the middle term.

3x² - 4x - 4 = 3x² - 6x + 2x - 4

Step 4:- Combine the first two terms and the last two terms, simplify the equation and take out any common numbers or expressions.

3x² - 6x + 2x - 4 = 3x (x - 2) + 2(x - 2)

Step 5:- Again take (x - 2) common from both the terms.

3x (x - 2) + 2(x - 2) = (x - 2) (3x + 2)

Therefore, (x - 2) and (3x + 2) are the factors of 3x² - 4x - 4.

Quadratic Trinomial in Two Variable

There is no specific way to solve a quadratic trinomial in two variables. Let's take an example.

Example: Factorize: x² + 3xy + 2y²

Solution

Step 1: These types of trinomials also follow the same rule as above, i.e., we need to break the middle term.

x² + 3xy + 2y² = x² + 2xy + xy + 2y²

Step 2: Simplify the equation and take out common numbers of expressions.

x² + 2xy + xy + 2y² = x (x + 2y) + y (x + 2y)

Step 3: Again take (x + 2y) common from both the terms.

x (x + 2y) + y (x + 2y) = (x + y) (x + 2y)

Therefore, (x + y) and (x + 2y) are the factors of x² + 3xy + 2y²

If Trinomial is an Identity

Let's see some algebraic identities that are mentioned in the table below:

Example: Factorize: 9x² + 12xy + 4y²

Solution:

Step 1: Identify which identity can be applied in the expression.

We can apply (x + y)² = x² + 2xy + y²

Step 2: Rearrange the expression so that it can appear in the form of the above identity.

9x² + 12xy + 4y² = (3x)² + 2 × 3x × 2y + (2y)²

Step 3: Once the expression is arranged in the form of the identity, write its factors.

(3x)² + 2 × 3x × 2y + (2y)² = (3x + 2y)² = (3x + 2y) (3x + 2y)

Therefore, (3x + 2y) is the factor of 9x² + 12xy + 4y².

Leading coefficient of 1

Let us look at an example.

Example: Factorize x² + 7x + 12

Solution:

Step 1: Compare the given equation with the standard form to obtain the coefficients.

ax² + bx + c is the standard form, comparing the equation x² + 7x + 12 we get a = 1, b = 7, and c = 12

Step 2: Find the paired factors of c i.e 12 such that their sum is equal to b i.e 7.

The pair factor of 12 are (1, 12), (2, 6), and (3, 4). Therefore, the suitable pair is 3 and 4.

Step 3: Add each number to x separately.

(x + 3) (x + 4)

Therefore, (x + 3) (x + 4) are the factors for x² + 7x + 12.

Factorizing with GCF

When the trinomial needs to be factorized where the leading coefficient is not equal to 1, the concept of GCF(Greatest Common Factor) is applied. Let us see the steps:

• Write the trinomial in descending order, from highest to lowest power.
• Find the GCF by factorization.
• Find the product of the leading coefficient 'a' and the constant 'c.'
• Find the factors of the product 'a' and 'c'. Pick a pair that sums up to get the number instead of 'b'.
• Rewrite the original equation by replacing the term “bx” with the chosen factors.
• Factor the equation by grouping.

Negative Terms

In some situations, a is negative, as in −ax² + bx + c. To make the factoring of the trinomial simpler, we factor out -1 from ax² as the first step and factor the rest of the expression. Let us look at an example.

Example: Factorize -4x² - 8x - 3.

Solution:

Step 1: Factor out -1 from the expression which changes the signs of the entire expression.

-1 (4x² + 8x + 3)

Step 2: Multiply the first term and the constant term.

4 × 3 = 12

Step 3: Break the middle term 8x such that on multiplying the resulting numbers, we get the result 12 (obtained from the previous step)

8x = 6x + 2x

6 × 2 = 12.

Step 4: Rewrite the middle term and group them.

-1 (4x² + 6x + 2x + 3)

-1 ([4x² + 2x] + [6x + 3])

Step 5: Factor the grouped terms.

-1 (2x[2x + 1] + 3[2x + 1])

Step 6: Write it as a binomial.

-1 [(2x + 1)(2x + 3)]

(-2x -1)(2x -3)

Therefore, (-2x -1) (2x -3) are the factors of -4x² - 8x - 3.

A trinomial can be a perfect square or a non-perfect square. We have two formulas to factorize a perfect square trinomial. But for factorizing a non-perfect square trinomial, we do not have any specific formula, instead, we have a process.

• The factoring trinomials formulas of perfect square trinomials are:

  a² + 2ab + b² = (a + b)²

  a² - 2ab + b² = (a - b)²

  For applying either of these formulas, the trinomial should be one of the forms a² + 2ab + b² (or) a² - 2ab + b².

• The process of factoring a non-perfect trinomial ax² + bx + c is:

  Step 1: Find ac and identify b.

  Step 2: Find two numbers whose product is ac and whose sum is b.

  Step 3: Split the middle term as the sum of two terms using the numbers from step - 2.

  Step 4: Factor by grouping.

To factorize a trinomial of the form ax² + bx + c, we can use any of the below-mentioned formulas:

• a² + 2ab + b² = (a + b)² = (a + b) (a + b)
• a² - 2ab + b² = (a - b)² = (a - b) (a - b)
• a² - b² = (a + b) (a - b)
• a³ + b³ = (a + b) (a² - ab + b²)
• a³ - b³ = (a - b) (a² + ab + b²)

Related Topics

Listed below are a few interesting topics that are related to factoring trinomials, take a look.

• Perfect Square Trinomial
• Quadratic Equation
• Binomial Distribution

What is Factoring Trinomials?

Factoring trinomials is the process of finding factors for a given trinomial expression. These factors are expressed in the form of binomials that are the sum and product of the terms in a trinomial. The general form of a trinomial is ax² + bx + c which is converted to a binomial in the form of (x + m)(x + n).

How Do You Factor Trinomials?

A trinomial can be factored in the form x² + bx + c. First, we need to find two integers (r and s) whose product sums up to c and through addition, it sums up to b. Once we find the two numbers, we rewrite the trinomial as x² + rx + sx + c and use the grouping and distributive property of the factor to find out the factors of the expression. The factors will be (x + r) (x + s).

What is the Formula to Factor a Trinomial?

The factoring trinomials formulas of perfect square trinomials are:

a² + 2ab + b² = (a + b)²

a² - 2ab + b² = (a - b)²

For applying either of these formulas, the trinomial should be one of the forms a² + 2ab + b² (or) a² - 2ab + b².

What are the Basic Rules to Factoring Trinomials?

The rules or points to remember while factoring a trinomial are:

• If all terms of the trinomial are positive, then all terms of the binomials will be positive.
• If the last term of the trinomial is negative but the middle term and the first term are positive, then one term of the binomial will be negative and the other will be positive. (The greater factor will be positive and the smaller will be negative).
• If the middle term and the last term of the trinomial are negative and the first term is positive, then the sign for one binomial will be positive and the other will be negative. (The greater factor will be negative and the smaller will be positive).
• If the last term and the first term of the trinomial are positive but the middle term is negative, then both signs of the binomials will be negative.
• Look for common factors for the trinomial ax2 + bx + c, where a is 1. First factor the common factor then factor the rest of the expression.
• If ax2 is negative in a trinomial, you can factor −1 out of the whole trinomial first.

What are the Algebraic Identities Used for Factoring Trinomials?

The three basic algebraic identities used in factorizing trinomials are:

• (x + y)² = x² + 2xy + y²
• (x - y)² = x² - 2xy + y²
• (x² - y²) = (x + y) (x - y)

How do you Factor a Perfect Square Trinomial?

A perfect square trinomial has three terms which may be in the form of (ax)²+ 2abx + b²= (ax + b)² (or) (ax)²−2abx + b² = (ax−b)².The steps to be followed to factor a perfect square polynomial are as follows.

• Verify whether the given perfect square trinomial is of the form a²+ 2ab + b² (or) a²−2ab + b².
• Check if the middle term is twice the product of the first and the third term. Also, check the sign of the middle term.
• If the middle term is positive, then compare the perfect square trinomial with a² + 2ab + b² and if the middle term is negative, then compare the perfect square trinomial with a² - 2ab + b².
• If the middle term is positive, then the factors are (a+b) (a+b) and if the middle term is negative, then the factors are (a-b) (a-b).