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Chapter 1: Problem 3

Perform the indicated operation and simplify. $$(3 a-4)(5 a+2)$$

Short Answer

Expert verified
15a^2 - 14a - 8

Step by step solution

01

Identify the Expression

The given expression is earlying expression is early equations, let's start with the equation: (3a - 4)(5a + 2).
02

Use the Distributive Property

To multiply the two binomials, apply the distributive property (also known as the FOIL method). Multiply each term in the first binomial by each term in the second binomial: 1. Multiply the first terms: ewline(3a) 脳 (5a) = 15a^22. Multiply the outer terms: (3a) 脳 (2) = 6a3. Multiply the inner terms: (-4) 脳 (5a) = -20a4. Multiply the last terms: (-4) 脳 (2) = -8
03

Combine Like Terms

Add together all the terms from the distributive property: (15a^2 + 6a - 20a - 8). Simplify by combining like terms (terms that have the same variable and exponent): 15a^2 + (6a-20a) - 8 = 15a^2 - 14a - 8.

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra. It helps in multiplying a single term and two or more terms inside a parenthesis. To apply this property, you need to 'distribute' the term outside the parenthesis to each term inside the parenthesis.
In our example:
  • Expression: \((3a - 4)(5a + 2)\)
  • Using the distributive property, we multiply each term in the first binomial by each term in the second binomial.
When we say 'distribute' in the given exercise, we perform the following operations:
  • Multiply \(3a\) by \(5a\).
  • Multiply \(3a\) by \(2\).
  • Multiply \(-4\) by \(5a\).
  • Multiply \(-4\) by \(2\).
This gives us the terms: \(15a^2, 6a, -20a, -8\).
Combining Like Terms
After distributing, the next step is to combine like terms. Like terms are terms that have the same variable and exponent.
  • For instance, \(6a\) and \(-20a\) are like terms because they both have the variable \(a\).
Here, we have the terms \(15a^2, 6a, -20a, -8\).
We combine \(6a\) and \(-20a\) since they are like terms. This simplification gives us:
\(6a - 20a = -14a\).
Hence, the expression now looks like this:
  • \(15a^2 - 14a - 8\)
FOIL Method
The FOIL method is a specific application of the distributive property used to multiply two binomials. FOIL stands for First, Outer, Inner, Last.
Here is how you use FOIL:
  • First: Multiply the first terms in each binomial: \(3a \times 5a = 15a^2\)
  • Outer: Multiply the outer terms in the binomials: \(3a \times 2 = 6a\)
  • Inner: Multiply the inner terms in the binomials: \(-4 \times 5a = -20a\)
  • Last: Multiply the last terms in each binomial: \(-4 \times 2 = -8\)
Finally, combine all these terms: \((15a^2 + 6a - 20a - 8)\)
Then, simply combine the like terms as described earlier, resulting in \(15a^2 - 14a - 8\).

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