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Chapter 6: Problem 187

Multiply. $$6 r(4 r+s)$$

Short Answer

Expert verified
24r^2 + 6rs

Step by step solution

01

Distribute the term outside the parentheses

Multiply the term outside the parentheses, which is 6r, by each term inside the parentheses (4r + s). The expression can be written as:6r(4r) + 6r(s)
02

Multiply the coefficients and variables

For the first term inside the parentheses: 6r(4r) = (6 * 4)(r * r) = 24r^2For the second term inside the parentheses: 6r(s) = 6rs
03

Combine the product terms

After distributing and multiplying the terms, combine them to get the final expression:24r^2 + 6rs

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Key Concepts

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

Distributive Property
When we multiply a polynomial by another polynomial or a monomial, we use the distributive property. This property states that every term inside the parentheses must be multiplied by the term outside. For instance, in the problem 6r(4r + s), we distribute 6r to both 4r and s. This ensures that every term in the expression is properly accounted for.

To apply the distributive property:
  • Identify the term outside the parentheses.
  • Multiply this term by each term inside the parentheses separately.
Using our exercise, we get:

6r(4r) + 6r(s).The multiplications are done individually and later combined.
Algebraic Expressions
An algebraic expression includes variables, coefficients, and constants. In the expression given in our problem, 6r(4r + s), 6r is the term outside the parentheses, and 4r + s is an algebraic expression inside.

Variables are symbols, like r and s, that represent numbers. Coefficients are numbers attached to variables, such as 6 in 6r and 4 in 4r. Constants are standalone numbers, but none are present in this specific exercise.
To effectively multiply algebraic expressions, ensure you correctly identify the components:
  • Coefficient (numerical factor)
  • Variable (like r or s)
  • Operator (such as + or -)
Combining these parts correctly leads to a proper solution.
Coefficient Multiplication
In polynomial multiplication, we often need to multiply coefficients and variables. Let's break this down with the steps from our exercise.
Step 1: Identify the coefficients and variables in each term.
For 6r(4r), 6 and 4 are coefficients, while r is the variable.

Step 2: Multiply the coefficients separately from the variables.
For 6r(4r): Coefficient multiplication: 6 * 4 = 24
Variable multiplication: r * r = r^2

When combined:
6r(4r) = 24r^2
  • Multiply the coefficients together.
  • Multiply the variables together.
Applying this to 6r(s), since there is no coefficient with s, only the multiplication with 6r is needed:
6r * s = 6rs.
By multiplying coefficients and variables correctly, we ensure the final expression is accurate.

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