Problem 80 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$3 p^{2}-6\left(5 p^{2}+p\right)+p^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(-26p^2 - 6p.\)

Step 1: Distribute through the Parentheses

First, distribute the \(-6\) to each term inside the parentheses\( (5p^2 + p) \). This gives:\[-6(5p^2) - 6(p) = -30p^2 - 6p.\]

Step 2: Combine All Like Terms

Now write down the expression with distributed terms:\[3p^2 - 30p^2 - 6p + p^2.\]Combine the like terms:- The terms with\( p^2 \) are:\( 3p^2 - 30p^2 + p^2 = (3 - 30 + 1)p^2 = -26p^2.\)- The term with\( p \) is:\(-6p.\)

Step 3: Write the Simplified Expression

The simplified form of the expression combining the like terms is: \[-26p^2 - 6p.\]

Key Concepts

Like Terms

In algebra, like terms are critical to simplifying expressions. They contain the same variable raised to the same power. This means you can only combine terms that have identical variable parts. For example, in the expression \(3p^2 - 30p^2 + p^2\), the terms \(3p^2\), \(-30p^2\), and \(p^2\) are like terms because they all contain \(p^2\). By combining these, we reduce the expression's complexity.To combine like terms, simply add or subtract the coefficients, which are the numerical parts of the terms. Here’s how it works:

• Take the coefficients: 3, -30, and 1.
• Add them: \(3 - 30 + 1 = -26\).
• The result is \(-26p^2\).

Remember, terms with different variables or powers cannot be combined, such as \(p^2\) and \(p\), which are distinct terms with different variable parts.

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions. It involves multiplying each term inside a bracket by the term outside. In our example, the expression is modified by distributing \(-6\) over each term inside the parentheses:\[-6(5p^2 + p) = -6 \times 5p^2 + (-6) \times p\].Here’s how this works in our expression:

• Multiply \(-6\) by \(5p^2\) to get \(-30p^2\).
• Multiply \(-6\) by \(p\) to get \(-6p\).

After distribution, our expression becomes \(-30p^2 - 6p\), which you can then combine efficiently with other like terms.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations (such as addition or multiplication). Simplifying these expressions often involves using the distributive property and combining like terms.Consider the expression: \(3p^2 - 6(5p^2 + p) + p^2\). This is a concise way to represent a mathematical relationship using:

• The variable \(p\), which stands for unknown values.
• Numerical coefficients like 3 and -6 that modify the variable terms.
• Operators like addition and subtraction.

To simplify such expressions, you should always begin by applying the distributive property where necessary, then proceed to combine any like terms. This process not only makes the expression easier to interpret but also shows the underlying relationships between variables and terms more clearly.