Factoring Polynomials with Three Terms Without a Greatest Common Factor (GCF)

Factoring a polynomial with three terms, but without a greatest common factor (GCF), can be approached using a method that involves identifying specific numerical properties of the polynomial. This article provides a detailed step-by-step guide to help you factor such polynomials effectively, with an example to illustrate the process.

Understanding the Polynomial Structure

A polynomial with three terms takes the form ( ax^2 bx c ), where ( a ), ( b ), and ( c ) are constants. When the polynomial does not have a GCF, you can still factor it using the grouping method. This technique relies on finding two numbers that satisfy specific criteria, which we will explore in detail.

Identifying the Terms

To begin, ensure that the polynomial is in the standard form:

[ ax^2 bx c ]

Identifying ( a ), ( b ), and ( c ) is crucial. For example, in the polynomial ( 2x^2 7x 3 ):

[ a 2 ] [ b 7 ] [ c 3 ]

Finding the Right Numbers

The next step involves finding two numbers that meet two specific criteria:

Multiplication: These two numbers must multiply to ( a times c ). Addition: They must also add up to ( b ).

For the polynomial ( 2x^2 7x 3 ), the product of the two numbers needs to be ( 2 times 3 6 ), and their sum must be ( 7 ). The numbers that satisfy these conditions are ( 6 ) and ( 1 ).

Why ( 6 ) and ( 1 )? Because: ( 6 times 1 6 ) ( 6 1 7 )

Rewriting the Middle Term

Next, rewrite the middle term (( bx )) into the sum of the two found numbers. For the polynomial ( 2x^2 7x 3 ), this becomes:

[ 2x^2 6x 1x 3 ]

Factoring by Grouping

Now, group the terms into two pairs:

[ (2x^2 6x) (1x 3) ]

Factor out the greatest common factor (GCF) from each pair:

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

Notice that both pairs share a common binomial factor, ( x 3 ).

Factoring Out the Common Binomial

Factor out the common binomial factor ( (x 3) ) from the expression:

[ (x 3)(2x 1) ]

This is the fully factored form of the polynomial ( 2x^2 7x 3 ).

Summary of the Process

To factor a polynomial with three terms and no GCF, follow these steps:

Identify the terms in the polynomial. Find two numbers that multiply to ( a times c ) and add to ( b ). Rewrite the middle term using the two found numbers. Group the terms and factor out the GCF from each pair. Factor out the common binomial factor.

Understanding and applying these steps will help you factor more complex polynomials with confidence.

Additional Methods for Factoring Trinomials

For certain polynomials, you may also consider using alternative methods:

Quadratic Formula: Useful for finding the roots of the polynomial directly, which can help with factoring. Grouping: Another method similar to the one described above, useful for specific structures of polynomials. Special Factorization: Techniques that apply to specific forms of polynomials. Trial and Error: Trying different combinations to factor the polynomial, especially useful for simpler polynomials.