Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((7 x-2)(4 x-3)\)

Step-by-Step Solution

Answer

The product of \((7x - 2)(4x - 3)\) is \(28x^2 - 29x + 6\).

Step 1: Distribute first term of the first binomial

First, distribute the term \(7x\) from the first binomial to each term in the second binomial, which yields \(28x^2\) and \(-21x\).

Step 2: Distribute second term of the first binomial

Next, distribute the term \(-2\) from the first binomial to each term in the second binomial. This results in \(-8x\) and \(6\).

Step 3: Combining like terms

Combine the middle terms \(-21x\) and \(-8x\), which yields \(-29x\). The final product of \((7x - 2)(4x - 3)\) simplifies to \(28x^2 - 29x + 6\).

Key Concepts

Binomials

In algebra, binomials are expressions that contain exactly two terms. These expressions are often used in polynomial multiplication problems. For example, in the exercise \((7x - 2)(4x - 3)\):

• "7x - 2" is the first binomial, consisting of the terms "7x" and "-2".
• Similarly, "4x - 3" is the second binomial, with the terms "4x" and "-3".

Binomials are a key concept in polynomial multiplication as they allow us to apply distribution methods to expand and simplify expressions.
Understand that when working with binomials, our goal is often to multiply and then simplify the expression into a polynomial, as will be shown through distribution.

Distribution

The distribution method, often termed the "distributive property", is essential for multiplying binomials. This technique involves multiplying each term in one binomial with every term in the other, ensuring no terms are overlooked.

Consider the binomials \((7x - 2)\) and \((4x - 3)\):

• First, take the term "7x" from the first binomial and distribute it over the terms in the second binomial. Multiply "7x" by "4x" to get \(28x^2\), and by "-3" to get \(-21x\).
• Next, distribute "-2" from the first binomial across the second binomial. Multiply "-2" by "4x" to get \(-8x\), and by "-3" to get 6.

Through this process, all combinations are covered, ensuring a comprehensive expansion of the binomials.

Combining Like Terms

Once distribution is complete, the next step involves organizing and simplifying the expression further. This is done by combining like terms. Like terms refer to terms that share the same variable and power. Let's see how this applies to our distributed expression.

After distributing, we have four terms: \(28x^2\), \(-21x\), \(-8x\), and 6.

• The middle terms, \(-21x\) and \(-8x\), are like terms because they both contain the term with the same power of x (\(x^1\)).
• Combine these like terms to simplify the expression to \(-29x\).

Completing this step, we can write the final simplified product of the original binomials as \(28x^2 - 29x + 6\). This step helps in reducing the expression to its simplest form.