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Chapter 5: Problem 24

Find each product. When possible, write down only the answer. $$(7 a-2 x)(x+a)$$

Short Answer

Expert verified
7a^2 - 2x^2 + 5ax

Step by step solution

01

- Apply the Distributive Property

To find the product of ull(7a - 2x)(x + a)ull, use the distributive property (also known as the FOIL method for binomials). Distribute each term in ull(7a - 2x)ull by each term in ull(x + a)ull.
02

- Distribute Each Term

First, distribute ull7aull:ull 7a(x + a) = 7ax + 7a^2Next, distribute ull-2xull:ull-2x(x + a) = -2x^2 - 2xaCombine these parts to get:7ax + 7a^2 - 2x^2 - 2xa
03

- Combine Like Terms

Combine the like terms ull7ax and ull-2xa). Remember, ull7axull - ull2xa simplifies to ull5ax. The entire expression then becomes:7a^2 - 2x^2 + 5ax.

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Key Concepts

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

Binomial Multiplication
When you need to multiply binomials like \( (7a - 2x)(x + a) \), think of each binomial as a group of two terms. The distributive property helps expand the product of these terms. Distribute each part of the first binomial to every part of the second binomial. This involves multiplying each term in the first group by each term in the second group. You'll end up with four separate products.

Remember, the process involves:
  • Multiplying the first term in the first binomial by both terms in the second binomial
  • Multiplying the second term in the first binomial by both terms in the second binomial
Once you have all the individual products, it's time to move on to the next step: combining like terms.
Combining Like Terms
Combining like terms is an essential step to simplify the expanded polynomial. Like terms have the same variable and exponent. For instance, in the expression obtained from the binomial multiplication: \( 7ax + 7a^2 - 2x^2 - 2xa \), the like terms are \( 7ax \) and \( -2xa \). Notice that these terms both contain \( ax \) with the same power.

When combining these terms, solve as follows:
  • Identify like terms, i.e., terms containing the same variables to the same power
  • Add or subtract the coefficients of these like terms
In our example, combining \( 7ax \) and \( -2xa \) results in: \( 7ax - 2xa = 5ax \). Thus, the simplified expression is \( 7a^2 - 2x^2 + 5ax \).

Keep practicing and identifying like terms will become easy and intuitive.
FOIL Method
The FOIL method is a specific technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, representing the order to multiply the terms:
  • First: Multiply the first terms of each binomial
  • Outer: Multiply the outer terms of each binomial
  • Inner: Multiply the inner terms of each binomial
  • Last: Multiply the last terms of each binomial
Consider our binomials \( (7a - 2x)(x + a) \):

1. First: \( 7a \times x = 7ax \)
2. Outer: \( 7a \times a = 7a^2 \)
3. Inner: \( -2x \times x = -2x^2 \)
4. Last: \( -2x \times a = -2xa \)

Follow these four steps and then combine the resulting terms: \( 7ax + 7a^2 - 2x^2 - 2xa \).
Finally, simplify as explained in the earlier sections to get \( 7a^2 - 2x^2 + 5ax \). The FOIL method is particularly handy as it ensures you don't miss any terms in the multiplication process.

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