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Chapter 10: Problem 191

Multiply the following binomials. Use any method. $$(x-4)(x+7)$$

Short Answer

Expert verified
The product is \(x^2 + 3x - 28\)

Step by step solution

01

- Write the binomials

The expression to multiply is given as ewline ewline \((x-4)(x+7)\)
02

- Apply the Distributive Property (FOIL Method)

Use the FOIL method to expand the binomials. FOIL stands for First, Outside, Inside, Last: ewline ewline \((x-4)(x+7)\) ewline ewline First: \((x \times x)\) \rightarrow \(x^2\) ewline ewline Outside: \((x \times 7)\) \rightarrow \(7x\) ewline ewline Inside: \((-4 \times x)\) \rightarrow \(-4x\) ewline ewline Last: \((-4 \times 7)\) \rightarrow \(-28\)
03

- Combine Like Terms

Combine the terms obtained from the FOIL method: ewline ewline \(x^2 + 7x - 4x - 28\) ewline ewline Combine the \(7x\) and \(-4x\) to get: ewline ewline \(x^2 + 3x - 28\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

FOIL method
When multiplying binomials, one effective approach is using the FOIL method. FOIL is an acronym that stands for First, Outside, Inside, Last. It helps you remember how to expand the multiplication of two binomials.

Let's take the example \((x-4)(x+7)\). We apply the FOIL method as follows:
  • First: Multiply the first terms in each binomial. Here, that's \x \times x = x^2\.
  • Outside: Multiply the outer terms. In our example, this is \x \times 7 = 7x\.
  • Inside: Multiply the inner terms: \-4 \times x = -4x\.
  • Last: Multiply the last terms in each binomial: \-4 \times 7 = -28\.
Putting all these together, we get \(x^2 + 7x - 4x - 28\).

The FOIL method effectively organizes our steps, ensuring all terms are considered for multiplication.
distributive property
The Distributive Property is a fundamental concept in algebra that applies when multiplying a single term by a binomial or polynomial. Essentially, it states that you must multiply each term in the binomial by the term outside the parentheses.

Using the exercise example, \((x-4)(x+7)\), let's look at how the distributive property works here. This method involves expanding each term of the first binomial by every term of the second binomial:
  • First, distribute \(x\) from \((x-4)\) to each term in \((x+7)\), resulting in \x \times x = x^2\ and \x \times 7 = 7x\.
  • Second, distribute \-4\ to each term in \((x+7)\), resulting in \-4 \times x = -4x\ and \-4 \times 7 = -28\.
The expanded form thus becomes \(x^2 + 7x - 4x - 28\).

By using the distributive property, you ensure that each term in one binomial multiplies every term in the other binomial.
like terms
After expanding binomials, you often get a combination of terms that can be combined. These are called 'like terms'. Like terms have the same variable raised to the same power.

In our example, we expanded \((x-4)(x+7)\) to get \(x^2 + 7x - 4x - 28\). Now, we should look for like terms to simplify:
  • Identify the like terms: \7x\ and \-4x\. Both terms have the variable \x\ to the first power.
  • Combine the like terms: Add \7x\ and \-4x\ to get \3x\.
So, after simplifying, our expression becomes \(x^2 + 3x - 28\).

Combining like terms is essential for simplifying expressions and getting to the final, most streamlined answer.

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