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 ax2+bx+cax2+bx+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 ax2+bx+cax2+bx+c. If $a=1$, the process is simpler. If a≠1a=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 $bx$ into two terms that can be factored.

Step 3: Split the Middle Term

Rewrite the middle term $bx$ as the sum of two terms using the numbers found in Step 2. For example, if you have x2+7x+12x2+7x+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 x2+3x+4x+12x2+3x+4x+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 x2+3x+4x+12x2+3x+4x+12, you group it as (x2+3x)+(4x+12)(x2+3x)+(4x+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 x2+7x+12x2+7x+12

1. Identify Coefficients: $a=1$, $b=7$, $c=12$.
2. Find Factors of $c$: Numbers that multiply to 12 and add up to 7 are 3 and 4.
3. Split Middle Term: x2+3x+4x+12x2+3x+4x+12.
4. Factor by Grouping: $x(x+3)+4(x+3)=(x+3)(x+4)$.

Thus, the factors of x2+7x+12x2+7x+12 are $(x+3)(x+4)$.

Example 2: Factoring x2+5x+6x2+5x+6

1. Identify Coefficients: $a=1$, $b=5$, $c=6$.
2. Find Factors of $c$: Numbers that multiply to 6 and add up to 5 are 2 and 3.
3. Split Middle Term: x2+2x+3x+6x2+2x+3x+6.
4. Factor by Grouping: $x(x+2)+3(x+2)=(x+2)(x+3)$.

Thus, the factors of x2+5x+6x2+5x+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 4×2−15x+94x2−15x+9

1. Factor Out Leading Coefficient: 4(x2−154x+94)4(x2−415​x+49​).
2. Identify New Coefficients: $a=1$, b=−154b=−415​, c=94c=49​.
3. Find Factors of $c$: Numbers that multiply to 9449​ and add up to −154−415​ are −32−23​ and $-3$.
4. Split Middle Term: 4(x2−32x−3x+94)4(x2−23​x−3x+49​).
5. Factor by Grouping: 4(x(x−32)−3(x−32))=4(x−32)(x−3)4(x(x−23​)−3(x−23​))=4(x−23​)(x−3).

Thus, the factors of 4×2−15x+94x2−15x+9 are 4(x−32)(x−3)4(x−23​)(x−3).

FAQ

Q: What is a trinomial?
A: A trinomial is a polynomial with three terms, typically in the form ax2+bx+cax2+bx+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

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.