Problem 48 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-4(4+2 p+5)\)

Step-by-Step Solution

Answer

\(-8p - 36\)

Step 1: Apply the Distributive Property

To remove the parentheses, apply the distributive property: multiply \(-4\) by each term inside the parentheses. The expression becomes:\(-4 \cdot 4 + (-4) \cdot (2p) + (-4) \cdot 5\).

Step 2: Multiply Each Term

Perform each multiplication separately:- \(-4 \cdot 4 = -16\)- \((-4) \cdot (2p) = -8p\)- \((-4) \cdot 5 = -20\)Now the expression is: \(-16 - 8p - 20\).

Step 3: Combine Like Terms

Combine the constant terms \(-16\) and \(-20\):\(-16 - 20 = -36\).This gives us the simplified expression:\(-8p - 36\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves rewriting expressions in a more concise form. The goal is to make the expression easier to work with, whether you are solving an equation or simply analyzing a problem. This often involves applying properties like the Distributive Property and combining like terms.

Simplification makes mathematical expressions more manageable and can reveal insights that are not immediately obvious in their original form. For example, when you simplify an expression, you highlight relationships between the variables and constants that were once hidden by parentheses or complex terms. This is why simplifying expressions is such an important step in solving algebraic problems efficiently.

In this particular exercise, "Simplifying Expressions" starts with the application of the distributive property:

• Multiply every term inside the parentheses by the factor outside.
• Remove the parentheses, so the expression is easier to read and manipulate.

Once you've applied these operations, simplifying further usually involves combining like terms, ensuring your final expression is as clear and concise as possible.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions. When you combine like terms, you add or subtract terms that have identical variable parts. This means they must have the same base and the same exponent. By combining them, you reduce the complexity of the expression and make it easier to manage.

In the step-by-step solution provided, the expression

• Combine the constant terms: - Combine \(-16 - 20 = -36\)

Visible simplification of terms occurs here, resulting in far simpler expressions which subsequently aids in problem-solving.By consistently using the strategy of combining like terms, you not only simplify expressions but lay the groundwork for solving more complex algebraic equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. These are the building blocks of algebra and are used to represent patterns, relationships, and equations in a concise form. Understanding how to manipulate these expressions is central to mastering algebra.

When working with algebraic expressions, you may encounter:

• Constants: Numerical values that do not change.
• Variables: Symbols that represent unknown or changeable values.
• Operations: Addition, subtraction, multiplication, and division.
• Coefficients: Numbers that multiply a variable.
• Terms: Components of the expression separated by addition or subtraction.

In our example exercise, algebraic expressions are organized through the use of the distributive property, enabling simplification by distributing the multiplication across each term.

This process aids in acquiring a clear, structured understanding of the operations involved, as well as their effects on the expression as a whole. As you progress in algebra, being adept at working with algebraic expressions is essential.