Problem 98 - Introductory Algebra for College Students

Question

Why is it a good idea to factor out the GCF first and then use other methods of factoring? Use \(3 x^{2}-18 x+15\) as an example. Discuss what happens if one first uses trial and error to factor as two binomials rather than first factoring out the GCF.

Step-by-Step Solution

Answer

Factoring out the GCF first simplifies the expression to a manageable size and lessens the risk of making errors. In contrast, direct trial and error can be more complex, especially with larger numbers or with expressions where a GCF is hidden, as illustrated by the quadratic expression \(3 x^{2}-18 x+15\).

Step 1: Factoring Out the GCF

Start by factoring out the greatest common factor (GCF) from the quadratic expression. The GCF of \(3 x^{2}-18 x+15\) is 3. This gives \(3(x^{2}-6x+5)\).

Step 2: Factoring the Quadratic Inside the Brackets

Now, we factor the quadratic expression inside the brackets. The factored form of \(x^{2}-6x+5\) is \((x-1)(x-5)\). So, the completely factored expression is \(3(x-1)(x-5)\).

Step 3: Trial and Error Without Factoring Out GCF

Without factoring out the GCF first, we try to factor the original equation directly. In this case, we aim to find a pair of numbers that multiply to give 15 (constant term) and add to give -18 (coefficient of x). But, no such pair of numbers exists due to the hidden GCF.

Step 4: Comparing the Approaches

Factoring out the GCF first simplifies the quadratic and makes the numbers smaller and manageable. This makes the rest of the factoring process easier and reduces the potential of making errors. On the other hand, direct trial and error may work in some cases but can be complicated and time-consuming when there is a hidden GCF