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Chapter 5: Problem 90

Find each product. \((3 r+5)(2 r+1)\)

Short Answer

Expert verified
The product is \[ 6r^2 + 13r + 5 \].

Step by step solution

01

- Apply the distributive property

Multiply each term in the first binomial \((3r + 5)\) by each term in the second binomial \((2r + 1)\).
02

- Perform the multiplication

Calculate \[ (3r) \times (2r) = 6r^2 \]. Then, calculate \[ (3r) \times 1 = 3r \]. Next, calculate \[ 5 \times (2r) = 10r \]. Lastly, compute \[ 5 \times 1 = 5 \].
03

- Combine like terms

Combine the results: \[ 6r^2 + 3r + 10r + 5 \]. Simplify by combining the like terms \[3r + 10r = 13r \]. The final expression is \[ 6r^2 + 13r + 5 \].

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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 essential for multiplying polynomials, especially binomials. It lets you simplify expressions where you need to multiply a single term by each term within a parenthesis. In this exercise, the distributive property is applied to multiply the terms in the binomial \(3r + 5\) by each term in \(2r + 1\). This breaks down the multiplication so you can handle it step by step. Let鈥檚 look closer:

  • Multiply \(3r\) by \(2r\) to get \[6r^2\]
  • Next, multiply \(3r\) by \(1\) to get \[3r\]
  • Then, multiply \(5\) by \(2r\) to get \[10r\]
  • Lastly, multiply \(5\) by \(1\) to get \[5\]
By handling each multiplication separately, the problem becomes much simpler. Afterward, combine the results to move to the next step.
Binomials
A binomial is an algebraic expression that contains exactly two terms. In our example, we work with the binomials \(3r + 5\) and \(2r + 1\). Each term in a binomial can be a number, a variable, or both. Understanding binomials is crucial because:

  • They frequently appear in algebraic problems.
  • Knowing how to manipulate them prepares you for more complex polynomial operations.
To multiply two binomials, such as in our exercise, apply the distributive property so each term in the first binomial multiplies every term in the second. This ensures that no combinations are missed, leading to a complete and accurate solution.
Combining Like Terms
Combining like terms is the process of simplifying an expression by adding or subtracting terms that have identical variable parts. For example, in the expression we derived \(6r^2 + 3r + 10r + 5\), we can combine \(3r\) and \(10r\) because they both have the same variable part \(r\).

This simplification step is vital because:

  • It makes the final expression more manageable and cleaner.
  • In our example, combining \(3r\) and \(10r\) gives us \[13r\], resulting in the simplified expression \[6r^2 + 13r + 5\]
Understanding how to combine like terms accurately helps you solve more complicated algebraic expressions efficiently. Always look for terms with the same variable and exponent to combine them.

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