Problem 72 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ d^{2}-7 d-6 d+42 $$

Step-by-Step Solution

Answer

(d - 7)(d - 6)

Step 1: Group the Terms

Group the terms into two pairs: \[ (d^{2} - 7d) \text{ and } (-6d + 42) \]

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Pair

In the first group \(d^{2} - 7d\), factor out \(d\): \[ d(d - 7) \] In the second group \(-6d + 42\), factor out \(-6\): \[ -6(d - 7) \]

Step 3: Factor Out the Common Binomial

Notice that \(d - 7\) is common in both groups. Factor that out: \[ d(d - 7) - 6(d - 7) = (d - 7)(d - 6) \]

Key Concepts

greatest common factor

The Greatest Common Factor (GCF) is the highest number that divides exactly into two or more numbers. In algebra, it extends to the highest common term or expression that can be factored out of a set of terms.

For example, in the expression \(d^{2} - 7d\), the GCF is \(d\) because both terms contain \(d\).
Factoring the GCF simplifies expressions and is often the first step in factoring by grouping.

To identify the GCF:

• Look at the coefficients (numbers in front of variables)
• Identify common variables and their lowest powers

This helps to break down and simplify the problem.

binomials

Binomials are algebraic expressions containing two terms. These terms can be numbers, variables, or both. For instance, \(d^{2} - 7d\) and \(-6d + 42\) are binomials.

When factoring binomials:

• Identify the terms involved
• Look for patterns or common factors

In our example, each binomial was factored separately to find the GCF.

By factoring out the GCF, these binomials turn into simpler expressions, which can eventually help in combining them using factoring techniques.

factoring techniques

Factoring techniques are methods used to rewrite expressions as products of simpler factors. One pivotal technique is factoring by grouping.

Here's a step-by-step process:

• Group the terms: In the example, terms were grouped as \( (d^{2} - 7d) \) and \(-6d + 42\).
• Factor the GCF from each group: For \(d^{2} - 7d\), the GCF is \(d\), leading to \(d(d - 7)\). For \(-6d + 42\), the GCF is \(-6\), leading to \(-6(d - 7)\).
• Factor out the common binomial: Both groups contain the binomial \(d - 7\). Factoring it out results in \( (d - 7)(d - 6) \).

These steps ensure a systematic way of simplifying expressions, making complex equations easier to handle.