Problem 76 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ 40 x^{2}-35 x-8 x+7 $$

Step-by-Step Solution

Answer

\((5x - 1)(8x - 7)\)

Step 1: Group the terms

Group the terms into two pairs: divide the expression into two parts: \( (40x^2 - 35x) - (8x - 7) \)

Step 2: Factor out the greatest common factor (GCF) from each group

Factor out the GCF from each grouped part: From the first group: \( 40x^2 - 35x = 5x(8x - 7) \)From the second group: \( -8x + 7 = -1(8x - 7) \)

Step 3: Factor out the common binomial

Notice that both groups now have a common binomial factor \( 8x - 7 \). Factor this out to get:\( (5x - 1)(8x - 7) \)

Key Concepts

Greatest Common Factor (GCF)

When we talk about the Greatest Common Factor (GCF), we refer to the largest number or algebraic expression that can evenly divide all terms in a particular group. This is crucial in the factor by grouping method.

In the example given, when we look at the expression \(40x^2 - 35x - 8x + 7\), we first group it into two parts, \(40x^2 - 35x\) and \(-8x + 7\).

Next, for each group, find the GCF:

• For the first group \(40x^2 - 35x\), the GCF is \(5x\).
• For the second group, \(-8x + 7\), the GCF is \(-1\).
  

By factoring out the GCFs, we simplify each group to read: \(5x(8x - 7)\) and \(-1(8x - 7)\).

Understanding how to find and use the GCF is fundamental in simplifying polynomials and in factoring by grouping.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as the product of its factors. This helps in solving for the roots or simplifying the expression further.

In the given example, we factor the polynomial \(40x^2 - 35x - 8x + 7\) by grouping. Initially, this expression doesn’t look like it can be easily factored. However, after grouping and factoring out the GCFs, it transforms into simpler binomial products.

Let’s break it down step-by-step:

• Step 1: Group the terms: \( (40x^2 - 35x) - (8x - 7) \).
• Step 2: Factor out the GCF from each group: \(5x(8x - 7)\) and \(-1(8x - 7)\).
• Step 3: Notice the common binomial factor: \(8x - 7\) and factor it out to get: \( (5x - 1)(8x - 7) \).

By following these steps, we can break down more complex polynomials into easily manageable factors.

Binomial Factors

A binomial is a polynomial with exactly two terms. When factoring polynomials especially by grouping, we often end up with common binomial factors.

In our exercise, after grouping and factoring out the GCFs, we identify the common binomial factor \(8x - 7\).

This is the crucial step that allows us to combine the expressions: \( (40x^2 - 35x) = 5x(8x - 7) \) and \( (-8x + 7) = -1(8x - 7) \) share the binomial factor \(8x - 7\).

Factoring out the binomial gives us the final expression: \[ (5x - 1)(8x - 7) \]

• Always look for common binomial factors after grouping and factoring out the GCF.
• This strategy makes complex polynomial expressions more approachable.

Understanding binomial factors and how to identify them is essential to mastering polynomial factoring by grouping.