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Chapter 4: Problem 39

Use the FOIL method to find each product. $$ (3 x-2)(5 x-1) $$

Short Answer

Expert verified
The product is \(15x^2 - 13x + 2\).

Step by step solution

01

- First Terms

Multiply the first terms in each binomial: \(3x \cdot 5x = 15x^2\)
02

- Outer Terms

Multiply the outer terms in each binomial: \(3x \cdot (-1) = -3x\)
03

- Inner Terms

Multiply the inner terms in each binomial: \((-2) \cdot 5x = -10x\)
04

- Last Terms

Multiply the last terms in each binomial: \((-2) \cdot (-1) = 2\)
05

- Combine Like Terms

Combine all the terms obtained from the previous steps: \[15x^2 - 3x - 10x + 2\]Combine the like terms \[-3x - 10x = -13x\]So the final expression is: \[15x^2 - 13x + 2\]

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Key Concepts

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

Binomial Multiplication
Whenever you're working with binomials, such as (3x-2) and (5x-1), you may need to multiply them together. This is called binomial multiplication. The FOIL method is a standard technique for this.
FOIL stands for First, Outer, Inner, Last, representing a sequence to follow:
  • First terms: Multiply the first terms of each binomial.
  • Outer terms: Multiply the outer terms.
  • Inner terms: Multiply the inner terms.
  • Last terms: Multiply the last terms.
By following these steps, you can systematically find the product of two binomials without missing any terms.

In our example: (3x-2)(5x-1), we start by applying this method step by step:

1. First terms: 3x * 5x = 15x^2
2. Outer terms: 3x * (-1) = -3x
3. Inner terms: -2 * 5x = -10x
4. Last terms: -2 * -1 = 2

Then, you combine all these together, giving you the full expression: 15x^2 - 3x - 10x + 2
Polynomial Expressions
When you multiply binomials, you often get a polynomial expression as a result. A polynomial is simply an algebraic expression with multiple terms.
In the example, (3x-2)(5x-1) produces a polynomial: 15x^2 - 3x - 10x + 2.

Here are some key points about polynomial expressions:
  • Terms: Each part of the expression separated by a plus or minus sign (like 15x^2, -3x, -10x, and 2).
  • Degree: The highest exponent of the variable (like x) in the polynomial.In our case, it鈥檚 2 (from 15x^2).
  • Coefficients: The numerical part of the terms (15 in 15x^2, -3 in -3x, etc.).

Understanding these parts is essential because it helps you manipulate and simplify the expressions more effectively.
Combining Like Terms
Once you have the expanded form of a polynomial, the next step is to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power.
For example, in 15x^2 - 3x - 10x + 2, the terms -3x and -10x are like terms because they both have the variable x raised to the first power.
Combining these gives:
-3x - 10x = -13x.

Combining like terms helps to simplify the expression to its most concise form.
After combining, our polynomial becomes 15x^2 - 13x + 2.
This makes it easier to understand and work with in future calculations.

Thus, the final simplified expression for (3x-2)(5x-1) is 15x^2 - 13x + 2.

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Most popular questions from this chapter

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The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$

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