How To Factor Polynomials With 3 Terms

How to Factor Polynomials with 3 Terms: A Comprehensive Guide

Factoring polynomials is a fundamental skill in algebra, and understanding how to factor polynomials with 3 terms (trinomials) is crucial for solving quadratic equations and other algebraic problems. This guide will walk you through the steps to factor trinomials, provide examples, and include a FAQ section to address common questions.

What is a Trinomial?

A trinomial is a polynomial with three terms, typically in the form $a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. The process of factoring involves breaking down this trinomial into simpler factors, usually in the form of binomials.

Steps to Factor Trinomials

Step 1: Identify the Coefficients

Identify the coefficients $a$ , $b$ , and $c$ in the trinomial $a x^{2} + b x + c$ . If $a = 1$ , the process is simpler. If $a \neq = 1$ , you will need to factor out the leading coefficient before proceeding.

Step 2: Find the Factors of $c$

Find two numbers whose product is $c$ and whose sum is $b$ . These numbers will help you split the middle term $b x$ into two terms that can be factored.

Step 3: Split the Middle Term

Rewrite the middle term $b x$ as the sum of two terms using the numbers found in Step 2. For example, if you have $x^{2} + 7 x + 12$ , you need to find numbers that multiply to 12 and add up to 7. The numbers are 3 and 4, so you rewrite it as $x^{2} + 3 x + 4 x + 12$ .

Step 4: Factor by Grouping

Group the terms into pairs and factor out the greatest common factor (GCF) from each pair. For the example $x^{2} + 3 x + 4 x + 12$ , you group it as $(x^{2} + 3 x) + (4 x + 12)$ and then factor out the GCF from each group: $x (x + 3) + 4 (x + 3)$ . Finally, factor out the common binomial factor: $(x + 3) (x + 4)$ .

Examples

Example 1: Factoring $x^{2} + 7 x + 12$

Identify Coefficients: $a = 1$ , $b = 7$ , $c = 12$ .
Find Factors of $c$ : Numbers that multiply to 12 and add up to 7 are 3 and 4.
Split Middle Term: $x^{2} + 3 x + 4 x + 12$ .
Factor by Grouping: $x (x + 3) + 4 (x + 3) = (x + 3) (x + 4)$ .

Thus, the factors of $x^{2} + 7 x + 12$ are $(x + 3) (x + 4)$ .

Example 2: Factoring $x^{2} + 5 x + 6$

Identify Coefficients: $a = 1$ , $b = 5$ , $c = 6$ .
Find Factors of $c$ : Numbers that multiply to 6 and add up to 5 are 2 and 3.
Split Middle Term: $x^{2} + 2 x + 3 x + 6$ .
Factor by Grouping: $x (x + 2) + 3 (x + 2) = (x + 2) (x + 3)$ .

Thus, the factors of $x^{2} + 5 x + 6$ are $(x + 2) (x + 3)$ .

Factoring Trinomials with Leading Coefficient Other Than 1

If the leading coefficient $a$ is not 1, you need to factor it out first before proceeding with the steps above.

Example: Factoring $4 x^{2} - 15 x + 9$

Factor Out Leading Coefficient: $4 (x^{2} - 4 15 x + 4 9)$ .
Identify New Coefficients: $a = 1$ , $b = - 4 15$ , $c = 4 9$ .
Find Factors of $c$ : Numbers that multiply to $4 9$ and add up to $- 4 15$ are $- 2 3$ and $- 3$ .
Split Middle Term: $4 (x^{2} - 2 3 x - 3 x + 4 9)$ .
Factor by Grouping: $4 (x (x - 2 3) - 3 (x - 2 3)) = 4 (x - 2 3) (x - 3)$ .

Thus, the factors of $4 x^{2} - 15 x + 9$ are $4 (x - 2 3) (x - 3)$ .

FAQ

Q: What is a trinomial?
A: A trinomial is a polynomial with three terms, typically in the form $a x^{2} + b x + c$ .

Q: How do I factor a trinomial?
A: To factor a trinomial, identify the coefficients, find two numbers whose product is $c$ and whose sum is $b$ , split the middle term, and then factor by grouping.

Q: What if the leading coefficient is not 1?
A: If the leading coefficient $a$ is not 1, factor it out first before proceeding with the steps to factor the trinomial.

Q: Can all trinomials be factored?
A: Not all trinomials can be factored into simpler binomials. Some may be irreducible over the real numbers.

Q: How do I check if my factoring is correct?
A: To check if your factoring is correct, multiply the factors back together to ensure you get the original trinomial.

Table: Key Steps and Examples

Step	Description	Example
Identify Coefficients	Identify $a$ , $b$ , and $c$ in $a x^{2} + b x + c$	$x^{2} + 7 x + 12$ : $a = 1$ , $b = 7$ , $c = 12$
Find Factors of $c$	Find numbers that multiply to $c$ and add up to $b$	$x^{2} + 7 x + 12$ : Numbers are 3 and 4
Split Middle Term	Rewrite $b x$ as the sum of two terms using the found numbers	$x^{2} + 3 x + 4 x + 12$
Factor by Grouping	Group terms and factor out the GCF from each group	$x (x + 3) + 4 (x + 3) = (x + 3) (x + 4)$

Additional Resources

For more detailed information on factoring polynomials, you can refer to the following Wikipedia article:

Polynomial: Polynomial – Wikipedia

This guide has provided a comprehensive overview of how to factor polynomials with 3 terms, along with examples and a FAQ section to address common questions. By following these steps, you can become proficient in factoring trinomials and solving quadratic equations.

How to Factor Polynomials with 3 Terms: A Comprehensive Guide

What is a Trinomial?

Steps to Factor Trinomials

Step 1: Identify the Coefficients

Step 2: Find the Factors of $c$

Step 3: Split the Middle Term

Step 4: Factor by Grouping

Examples

Example 1: Factoring $x^{2} + 7 x + 12$

Example 2: Factoring $x^{2} + 5 x + 6$

Factoring Trinomials with Leading Coefficient Other Than 1

Example: Factoring $4 x^{2} - 15 x + 9$

FAQ

Table: Key Steps and Examples

Additional Resources

Comments

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How to Factor Polynomials with 3 Terms: A Comprehensive Guide

What is a Trinomial?

Steps to Factor Trinomials

Step 1: Identify the Coefficients

Step 2: Find the Factors of cc

Step 3: Split the Middle Term

Step 4: Factor by Grouping

Examples

Example 1: Factoring x2+7x+12x2+7x+12

Example 2: Factoring x2+5x+6x2+5x+6

Factoring Trinomials with Leading Coefficient Other Than 1

Example: Factoring 4×2−15x+94x2−15x+9

FAQ

Table: Key Steps and Examples

Additional Resources

Comments

Leave a Reply Cancel reply

Step 2: Find the Factors of $c$

Example 1: Factoring $x^{2} + 7 x + 12$

Example 2: Factoring $x^{2} + 5 x + 6$

Example: Factoring $4 x^{2} - 15 x + 9$