Factoring Trinomials | Calculator with various steps, examples & solutions

The Factoring Trinomials Calculator is a free online calculator that shows the factors of a particular trinomial. However, a trinomial is a polynomial with only three terms in mathematics. Variables, constants, and mathematical operators are examples. When you factor trinomials, you’ll get two binomials. However, when these two binomial terms are multiplied, a particular trinomial is obtained. A trinomial can be of any shape, such as a perfect square trinomial or a difference of squares.

A trinomial is, for example, x²+x – 6.

The resultant two binomials are (x-2) and (x+3) when it is factored in.

Factoring Trinomials Calculator with Steps

The following is how to use the factoring trinomials calculator:

Step 1:

In the input field, type the trinomial function.

Step 2:

Meanwhile, to retrieve the result, click the “FACTOR” button.

Step 3:

However, in the new window, the factors of a trinomial will be presented.

Factoring Trinomials meaning

A trinomial is a polynomial with three terms. X² + bx + c is a common (but not always!) form for trinomials. However, factoring trinomials may appear challenging at first glance, but you can use some interesting mathematical patterns to factor even the most complex trinomials. However, expanding an equation into the product of two or more binomials is called factoring a trinomial. (x + m) (x + n) is how it’s written. Meanwhile, a trinomial can be factored in a number of different ways.

Factoring Trinomials Calculator with Solution

Let’s discuss each case.

Factoring Quadratic Trinomial in One Variable

However, in one variable, the general version of the quadratic trinomial formula is ax² + bx + c, where a, b, and c are constant terms and none of a, b, or c is zero. If b² – 4ac > 0, we can factorize a quadratic trinomial for any value of a, b, and c. So it means that a(x + h)(x + k), where h and k are real values, equals a(x + h)(x + k). Let’s look at an example of how to factor a quadratic trinomial.

Example:

Factorize: 3x² – 4x – 4

Solution:

Step 1: Multiply the x² coefficient with the constant term first.

3 × -4 = -12

Step 2: Meanwhile, multiply the resulting integers to achieve the result -12 by breaking the middle term -4x (obtained from the first step).

-4x = -6x + 2x

-6 × 2 = -12

Step 3: Using the modification in the middle term, rewrite the main equation.

3x² – 4x – 4 = 3x² – 6x + 2x – 4

Step 4: However, combine the first and second terms, simplify the equation, and eliminate any common numbers or phrases.

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

Step 5: Meanwhile, take the (x – 2) common phrase from both terms once more.

3x (x – 2) + 2(x – 2) = (x – 2) (3x + 2)

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

Factoring Quadratic Trinomial in Two Variable

In two variables, there is no precise strategy to solve a quadratic trinomial. However, let’s have a look at an example.

Example:

Factorize: x² + 3xy + 2y²

Solution:

Step 1: However, these forms of trinomials likewise follow the same rule as the previous ones, namely, that the middle term must be broken.

X² + 3xy + 2y² = x² + 2xy + xy + 2y²

Step 2: Meanwhile, remove common numbers of expressions from the equation to make it easier to understand.

X² + 2xy + xy + 2y² = x (x + 2y) + y (x + 2y)

Step 3: Take the common phrase (x + 2y) from both terms.

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

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

Factoring Trinomials with GCF

When factoring a trinomial with a leading coefficient that is not equal to 1, the notion of GCF (Greatest Common Factor) is used.

Let’s take a look at the steps:

Write the trinomial from highest to lowest power in decreasing order.
However, factorization is used to find the GCF.
The product of the leading coefficient ‘a’ and the constant ‘c’ has to be found.
Then look for the product ‘a’ and ‘c’ factors. Instead of ‘b,’ choose a pair that adds up to get the number.
Meanwhile, replace the phrase “bx” with the chosen factors in the original equation.
By grouping, factor the equation.

Factoring Trinomials Steps

A perfect square or a non-perfect square is a trinomial. A perfect square trinomial can be factored using two formulas. However, we don’t have a specific formula for factoring a non-perfect square trinomial, but we do have a process.

Meanwhile, the perfect square trinomials factoring trinomials formulas are as follows:

a² + 2ab + b² = (a + b)²

a² – 2ab + b² = (a – b)²

The trinomial should be one of the forms when using either of these formulas.

a² + 2ab + b² (or) a² – 2ab + b².

Factoring a non-perfect trinomial ax² + bx + c is as follows:

Step 1: Locate ac and determine b.
Find two numbers whose product is ac and whose sum is b in Step 2.
Step 3: Using the numbers from step 2, split the middle phrase into two terms.
Step 4: Factoring by grouping is the final step.

Factoring Trinomials Formula

Any of the formulas below can be used to factorize a trinomial of the form ax² + bx + c:

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)

Factor trinomials with a leading coefficient of 1

Factoring Trinomials: x² + bx + c

X² + bx + c trinomials are frequently factored as the product of two binomials. However, a binomial is a two-term polynomial, so keep that in mind. Let’s start by looking at what happens when two binomials are multiplied, such as (x + 2) and (x + 5).

Example:

Problem: Multiply (x + 2)(x + 5).

(x + 2)(x + 5)

To multiply binomials, use the FOIL method.

X² + 5x + 2x +10

Then combine phrases that are similar, such as 2x and 5x.

Answer = x² + 7x +10

Factorize Trinomials By Splitting The Middle Term

Problems with Solutions

Factorization of quadratic polynomials in the form x² + bx + c (Type I).

(i) We must find numbers p and q such that p + q = b and pq = c in order to factorize x² + bx + c.

(ii) However, after determining p and q, we split the middle term in the quadratic as px + qx and group the terms to obtain the appropriate components.

Example 1:

Each of the following expressions should be factored:

(i) X² + 6x + 8 (ii) x² + 4x –21

Solution:

(i) We select two values p and q such that p + q = 6 and pq = 8 to factorize x² + 6x + 8.

2 + 4 = 6 and 2 4 = 8 are obvious.

In the following quadratic, we can partition the middle term 6x into 2x + 4x, so that

x²+ 6x + 8 = x2 + 2x + 4x + 8

= (x2 + 2x) + (4x + 8)

= x (x + 2) + 4 (x+ 2)

= (x + 2) (x + 4)

(ii) We must find two values p and q such that p + q = 4 and pq = – 21 in order to factorize x² + 4x – 21.

7 + (– 3) = 4 and 7 – 3 = – 21 are obvious.

We’ve now divided the middle term into 4x of x² + 4x – 21 as 7x – 3x, so that

X² + 4x – 21 = x² + 7x – 3 x – 21

= (x² + 7x) – (3x + 21)

= x (x + 7) – 3 (x + 7) = (x + 7) (x – 3)

Example 2:

Factor each of the quadratic polynomials below: x² – 21x + 108

Solution:

To factorize x² – 21x + 108, we must find two numbers whose total is – 21 and whose product is 108.

Clearly, – 21 = – 12– 9 and – 12 × – 9 = 108

x² – 21 x + 108 = x² – 12 x – 9x + 108

= (x² – 12 x) – (9x– 108)

= x(x – 12) – 9 (x – 12) = (x–12) (x – 9)

Factor trinomials with a leading coefficient other than 1

Factoring Trinomials: ax² + bx + c

ax² + bx + c is the generic form of trinomials with a leading coefficient of a. As you can see above, the factor of a can sometimes be factored; this occurs when a can be factored out of all three terms. The remaining trinomial will thus be easier to factor, as the leading term will only be an x² term rather than an ax² term.

However, if the coefficients of all three parts of a trinomial do not have a common factor, the trinomial must be factored using a coefficient other than 1.

Trinomials of the type ax² + bx + c are factored.

Find two integers, r and s, whose sum is b and whose product is ac to factor a trinomial of the type ax² + bx + c. Meanwhile, rewrite the trinomial as ax² + rx + sx + c, then factor the polynomial using grouping and the distributive property.

However, factoring trinomials in the form x² + bx + c is nearly the same as factoring trinomials in the form a = 1. You’re seeking for two elements whose sum is b and whose product is a • c.

Factoring trinomials Calculator box method

Factoring a trinomial of the form ax² + bx + c using the box approach is probably the best method.

We use a 2x² square to put the terms of the trinomial in the box.

Example:

Factor 2x² + 9x + 10 using the box method.

Step 1

To begin, enter 2x² and 10 in the boxes below in the order displayed. However, the first term will always go in the first row and first column, while the last term will go in the second row and second column.

Step 2

Meanwhile, multiply the first term by the last term to get the following result: 2x² x 10 = 20x²

Look for 20x² variables that add up to 9x in the second term.

20x² = 1x × 20x

Now, 20x² = 2x × 10x

20x² = 4x × 5x

Put 4x and 5x in the box since they total up to 9x.

Step 3

Meanwhile, each row’s highest common component should be noted on the left or right side of the box. Then, however, for each column, determine the greatest common factor and write it on the top or bottom of the box.

A common pitfall to avoid when factoring using the box method

When factoring with the box technique, make sure the greatest common factor of a, b, and c in ax² + bx + c equals 1, as shown in example #2.

Meanwhile, using the box approach, multiply 6x² + 27x + 30.

Step 1

First, put 6x² and 30 in the box below as shown.

Step 2

Meanwhile, multiply the first term by the last term to get the following result: 180x² = 6x² x 30

However, look for 180x² factors, which sum up to 27x.

Put 12x and 15x in the box because 12x + 15x = 27x and 12x + 15x = 180x².

Step 3

Each row’s highest common component should be noted on the left or right side of the box. Then, however, for each column, determine the greatest common factor and write it on the top or bottom of the box. As a result, we have the following box.

We do, however, have a minor issue. 15 times 3x is not equal to 15x, and 15 times 6 is not equal to 30. 6x times 3x is not equal to 6x², 6x times 6 is not equal to 12x, 6x times 6 is not equal to 12x, and 6x times 6 is not equal to 12x.

To understand what happened here, we must first make a critical remark.

Notice that 6x² + 27x + 30 = 3(2x² + 9x + 10 ) and we already factored 2x² + 9x + 10

2x² + 9x + 10 = (2x + 5)× (x + 2)

Therefore, 6x² + 27x + 30 = 3(2x² + 9x + 10 ) = 3(2x + 5)×(x + 2)

Meanwhile, we can factor the entire thing by 3 if we look at the box directly above again. And if we do that, we’ll receive the same result as seen below.