Problem 23 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (3 x+5)(2 x+1) $$

Step-by-Step Solution

Answer

The product of the two binomials \( (3x + 5)(2x + 1) \) is \( 6x^2 + 13x + 5 \)

Step 1: Distribution

Distribute each term of the first binomial, \(3x+5\), to every term of the second binomial, \(2x+1\). This results in four terms: \( (3x \cdot 2x) + (3x \cdot 1) + (5 \cdot 2x) + (5 \cdot 1) \)

Step 2: Simplification of Terms

Simplify each term from the previous step. This results in: \( 6x^2 + 3x + 10x + 5 \)

Step 3: Combination of Like Terms

Combine like terms (terms that have the same variable with the same exponent) to simplify the expression. In this case, combine the terms \(3x\) and \(10x\), which gives the final expression: \( 6x^2 + 13x + 5 \)

Key Concepts

Understanding Binomials

A binomial is a polynomial with exactly two terms. In the exercise, the expression is \(3x + 5\), which is a binomial because it has two distinct terms combined by a plus sign. Binomials can include numbers and variables, like \(3x\) and \(5\), with numbers representing coefficients and variables representing unknown values.

• The first term \(3x\) is called a 'linear term,' since it involves a variable raised to the first power.
• The second term \(5\) is a 'constant term,' since it doesn't include a variable.

When multiplying two binomials, such as \((3x + 5)\) and \((2x + 1)\), each term in one binomial must be multiplied by each term in the other. This is done through a procedure called distribution.

The Distribution Process

Distribution is a technique used to multiply each term in one binomial by each term in another to create a full expression. Imagine spreading the terms across each other, which is why it's called distribution. To multiply two binomials like \((3x + 5)(2x + 1)\), you follow these steps:

• Multiply the first term in the first binomial, \(3x\), by each term in the second binomial, \(2x\) and \(1\).
• Then multiply the second term of the first binomial, \(5\), by each term in the second binomial, \(2x\) and \(1\).

This results in four products, also known as 'partial products': - \(3x \cdot 2x = 6x^2\)- \(3x \cdot 1 = 3x\)- \(5 \cdot 2x = 10x\)- \(5 \cdot 1 = 5\)
Distribution allows us to methodically and systematically expand the product of two binomials into four separate terms.

Combining Like Terms Simplified

After distribution, you often end up with several terms in your polynomial expression. Many of these terms might be similar, known as 'like terms.' Like terms have the same variable with the same exponent.
For instance, after distributing \((3x + 5)(2x + 1)\), you get the expression \(6x^2 + 3x + 10x + 5\). To simplify, we need to combine the like terms, which here are the 'linear terms' \(3x\) and \(10x\).

Add the coefficients of the like terms together while maintaining the variable, which simplifies the middle terms to \(13x\).

The simplified polynomial expression is then \(6x^2 + 13x + 5\).

Combining like terms helps in making the polynomial much simpler and more manageable.