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Factoring Trinomials

October 31, 2023
written by Rida Mirza

Trinomial is a mathematical expression consisting of three terms. The terms are often made up of variables, coefficients, and constants. Trinomials can be written in the form: axÂ² + bx + c, where a, b, and c are numbers or algebraic expressions.

In this article, we will discuss how to factoring trinomials.

Standard Form

The standard form is,

axÂ² + bx + c

where a, b, and c are numbers or algebraic expressions

Factoring a Simple Trinomial

1: Identify Common Factors

Let’s consider the trinomial xÂ² + 5x + 6. First, look for any common factors among the three terms. In this case, there are none.

2: Split the Middle Term

To factor, we need to split the middle term (5x) into two terms that, when multiplied, give us the last term (6) and, when added, give us the middle term (5x).

We’re looking for two numbers that fit this criterion. In this example, it’s 2x and 3x because 2x * 3x = 6xÂ² and 2x + 3x = 5x.

3: Factor by Grouping

xÂ² + 2x + 3x + 6

Factor the common terms in each group:

x(x + 2) + 3(x + 2)

4: Final Step

You’ve got two expressions, x(x + 2) and 3(x + 2). Notice they both have the common factor of (x + 2). You can now factor that out:

(x + 2)(x + 3)

That’s the factored form of the trinomial!

Factoring a Trinomial with a Leading Coefficient

Now, let more complex trinomial: 3xÂ² + 10x + 7.

1: Identify Common Factors

Again, start by looking for common factors. In this case, there are none.

2: Split the Middle Term

We need to split the middle term (10x) into two terms that, when multiplied, give us the last term (7) and, when added, give us the middle term (10x). It’s 1x and 9x because 1x * 9x = 9xÂ² and 1x + 9x = 10x.

3: Factor by Grouping

Group the terms:

3xÂ² + 1x + 9x + 7

Factor the common terms in each group:

x(3x + 1) + 7(3x + 1)

4: Final Step

Both expressions have the common factor (3x + 1). Factor it out:

(3x + 1)(x + 7)

FAQs

What’s the importance of factoring trinomials?

Factoring trinomials is crucial for simplifying complex mathematical expressions and solving equations. It’s a fundamental skill in algebra and has applications in various fields, including physics and engineering.

Can all trinomials be factored?

Not all trinomials can be factored using integers or real numbers. Some might involve complex numbers or irreducible factors. However, many trinomials can be factored using the techniques explained here.

How can I check if I factored a trinomial correctly?

You can check your factoring by multiplying the factors to see if they equal the original trinomial. If they do, you’ve factored it correctly. Additionally, you can use the distributive property to verify.

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