Question

Factor by grouping. $$ 5 x^{2}-15 x+x-3 $$

Short answer

(x - 3)(5x + 1)

Step-by-step solution

Group the terms

Begin by grouping the terms in pairs to make it easier to factor. Group the first two terms together and the last two terms together: 5x^2 - 15x + x - 3 becomes (5x^2 - 15x) + (x - 3)

Factor out the greatest common factor (GCF) in each group

Factor out the greatest common factor from each group of terms. For the first group (5x^2 - 15x), factor out 5x. For the second group (x - 3), factor out 1: 5x(x - 3) + 1(x - 3)

Factor out the common binomial factor

Notice that (x - 3) is a common factor in both terms. Factor out (x - 3): (x - 3)(5x + 1)

Key concepts

Greatest Common Factor

The Greatest Common Factor, or GCF, is the largest number that divides exactly into two or more numbers. When factoring polynomials, finding the GCF is crucial. For example, in the polynomial \(5x^2 - 15x\), the GCF is \(5x\) because \(5x\) is the largest term that can divide both \(5x^2\) and \(15x\) without leaving a remainder. Recognizing and extracting the GCF from polynomial terms simplifies the expression and makes further factoring steps more manageable. To isolate the GCF, look at both coefficients and variables:

• Coefficients: Find the largest number that can evenly divide all the terms.
• Variables: Choose the variable with the smallest exponent that appears in each term.

Once you've identified the GCF, you can factor it out, reducing the polynomial to a simpler form.

Factoring Polynomials

Factoring a polynomial means breaking it down into simpler terms (or 'factors') that, when multiplied together, give the original polynomial. When you factor polynomials, you're essentially reversing the process of multiplication. In our example, to factor \(5x^2 - 15x + x - 3\), we use a method called 'grouping'. First, you group terms to find common factors:
1. \(5x^2 - 15x\)
2. \(x - 3\)Next, factor out the GCF from each group:

• For \(5x^2 - 15x\), the GCF is \(5x\), yielding \( \text{5x}(x - 3) \).
• For \(x - 3\), the GCF is \(1\), yielding \(1(x - 3) \).

Now, you look for a common binomial factor. In this case, \(x - 3\) is common to both groups.

Group Terms

Grouping terms is a useful technique in factoring, especially when dealing with polynomials that have four or more terms. The main idea is to rearrange and group terms into pairs or sets where common factors are more apparent. For the polynomial \(5x^2 - 15x + x - 3\):
1. First, group the terms: \((5x^2 - 15x) + (x - 3)\).2. Then, factor out the GCF from each group:

• From \(5x^2 - 15x\), factor out \(5x\): \(5x(x - 3)\).
• From \(x - 3\), factor out \(1\): \(1(x - 3)\).

Notice that both groups contain the common binomial factor \(x - 3\). You can now factor \(x - 3\) out: \((x - 3)(5x + 1)\). Grouping terms in this way helps you identify patterns and simplifies the factoring process.