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Chapter 15: Problem 65

Multiply. $$ 2(4 a+6 b-3 c) $$

Short Answer

Expert verified
8a + 12b - 6c

Step by step solution

01

Understand the Expression

Identify the given expression to multiply, which is 2(4a + 6b - 3c). This represents a single value (2) being multiplied by a polynomial inside the parentheses.
02

Distribute the 2

Multiply each term inside the parentheses by 2. Use the distributive property of multiplication over addition: 2 * 4a + 2 * 6b - 2 * 3c.
03

Perform the Multiplication

Multiply the constants with the variables: 2 * 4a = 8a, 2 * 6b = 12b, 2 * -3c = -6c.
04

Combine the Results

Combine the multiplied terms into a single expression: 8a + 12b - 6c.

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Key Concepts

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

polynomials
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can have constants, variables, and exponents that are non-negative integers.

In the given exercise, we have a polynomial inside the parentheses: \(4a + 6b - 3c\). This polynomial has three terms:
  • 4a (where 4 is the coefficient and a is the variable)
  • 6b (where 6 is the coefficient and b is the variable)
  • -3c (where -3 is the coefficient and c is the variable)
Each term has a coefficient and a variable. When working with polynomials, it's essential to recognize each term and its components. This helps in performing operations like addition, subtraction, and multiplication accurately.
multiplication
Multiplication is one of the fundamental arithmetic operations. In algebra, we often use multiplication to scale polynomials.

In this exercise, we use the distributive property to multiply 2 with every term inside the parentheses:
  • First, we multiply 2 by the first term (4a): \(2 * 4a = 8a\)
  • Next, we multiply 2 by the second term (6b): \(2 * 6b = 12b\)
  • Finally, we multiply 2 by the third term (-3c): \(2 * -3c = -6c\)
Thus, each term inside the parentheses is scaled by 2. This step-by-step multiplication makes the calculation more manageable, ensuring each term is correctly scaled.
algebraic expressions
An algebraic expression is a combination of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

In our exercise, \(2(4a + 6b - 3c)\) is an algebraic expression where we use multiplication over a polynomial. The final simplified expression, \(8a + 12b - 6c\), is also an algebraic expression.

To work with algebraic expressions efficiently, keep these tips in mind:
  • Identify and separate each term.
  • Perform operations step-by-step, especially when using properties like distributive property.
  • Combine like terms if needed.
By following these principles, you can simplify and understand complex algebraic expressions easily.

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Most popular questions from this chapter

Combine like terms. $$ 3 a-4 b-5 b-6 a $$

Solve each system using the substitution method. $$ \begin{aligned} &2 x-3=y\\\ &y-2 x=1 \end{aligned} $$

Translate each problem to an equation. Do not solve. The average U.S. citizen consumes 296 servings of Coca-Cola each year. This is 7.4 times the international average. What is the international average per capita consumption of Coke?

Solve each system using the elimination method. $$ \begin{aligned} &3 x+y=-1\\\ &2 x-3 y=-8 \end{aligned} $$

Remove parentheses and simplify. $$ 7-4[2(3-2 x)-5(4 x-3)] $$
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