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Chapter 5: Problem 3

Multiply. $$ 12(4 x-3) $$

Short Answer

Expert verified
The result is \( 48x - 36 \).

Step by step solution

01

Distribute 12 to each term

Multiply 12 by each term in the parentheses separately. Start with the first term: \[ 12 imes 4x = 48x \] Then multiply 12 by the second term: \[ 12 imes (-3) = -36 \]
02

Combine the results

Write down the results from the previous step to get the expression: \[ 48x - 36 \] There are no like terms to combine in this expression.

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental principle in algebra that simplifies expressions and equations. It involves multiplying a single term, often outside parentheses, by each term inside the parentheses. This process effectively "distributes" the multiplication across all terms within the parentheses.
For example, in the exercise given:
  • We have the expression \( 12(4x - 3) \).
  • Using the distributive property, 12 is multiplied with both \( 4x \) and \( -3 \).
This results in separate multiplication calculations: \( 12 \times 4x \) and \( 12 \times -3 \).
Starting with such multiplications helps break down complex expressions into more manageable parts.
Basics of Multiplication of Terms
Multiplying terms in algebra involves working with both numbers and variables, separately applying multiplication rules. When multiplying, follow these steps:
  • The numerical part: Simply multiply the numbers. For example, \( 12 \times 4 = 48 \). This gives us \( 48x \) as we multiply a number with a term including a variable.
  • The variable part: If variables were involved with powers, you would exponentiate based on the rules (e.g., \( x^a \times x^b = x^{a+b} \)), though in our exercise the variables were single powers and did not require additional power adjustments.
Continuing the process with \( 12 \times -3 \), focus solely on the numerical outcome, resulting in \(-36\).
Mastering multiplication of terms like this reinforces understanding of how numerical values and variables interact through arithmetic operations.
Combining Like Terms
Combining like terms in algebra involves simplifying expressions by merging terms with the same variables and exponents. This simplifies the equation and makes it easier to manage.
For the expression we derived, \( 48x - 36 \), each term is already in its simplest form, as there are no similar terms to combine. When an expression has like terms:
  • You add or subtract the coefficients (numbers in front of the terms) while keeping the common variable and exponent unchanged.
  • For instance, if you had \( 5x + 3x \), you would combine to get \( 8x \).
In our specific case, with \( 48x - 36 \), it's crucial to recognize that further simplification isn鈥檛 possible, highlighting the simplest outcome from using the distributive property and multiplication.

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