Problem 101 - Intermediate Algebra

Question

Simplify each expression. $$21\left(\frac{6}{7} h^{2}-\frac{15}{21} h\right)+21\left(\frac{1}{3} h\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \(18h^{2} - 8h\).

Step 1: Distribute the First Term

First, let's distribute the 21 in the term \(21\left(\frac{6}{7} h^{2}-\frac{15}{21} h\right)\). This is done by multiplying 21 with each term inside the parenthesis.1. \(21 \times \frac{6}{7} h^{2} = \frac{126}{7} h^{2} = 18h^{2}\)2. \(21 \times -\frac{15}{21} h = -15h\)This simplifies to \(18h^{2} - 15h\).

Step 2: Distribute the Second Term

Next, distribute the 21 in the term \(21\left(\frac{1}{3} h\right)\) by multiplying 21 by \(\frac{1}{3} h\).1. \(21 \times \frac{1}{3} h = \frac{21}{3} h = 7h\)This simplifies to \(7h\).

Step 3: Combine Like Terms

Add the results from the first and second terms: \(18h^{2} - 15h + 7h\).1. Combine \(-15h\) and \(7h\): \(-15h + 7h = -8h\)So, the expression becomes \(18h^{2} - 8h\).

Key Concepts

Distribution in Algebra

When dealing with algebraic expressions, the principle of distribution is essential. The distributive property allows us to multiply a single term by each term within a parenthesis. This is a key step in simplifying algebraic expressions.
This property states that for any numbers or expressions \( a \), \( b \), and \( c \), the expression \( a(b + c) \) can be expanded to \( ab + ac \).

Here's how it applies in practical situations:

• Multiply each term inside the bracket by the number outside. This step ensures each component of the expression is correctly proportioned by the multiplier.
• Remember to keep track of the signs (positive or negative) for each term involved.

In our given problem, the number \( 21 \) is distributed over two different sets of terms in brackets, working through each term separately. It helps to ensure every term is treated correctly and individually multiplied by \( 21 \).

Simplifying Polynomials

Simplifying polynomials involves making expressions more manageable by reducing complexity without changing their value. Often, we combine steps such as distribution, eliminating fractions, and performing basic arithmetic.

• First, apply the distribution to eliminate parentheses as often seen in algebraic simplification.
• Next, look for opportunities to simplify fractions and perform any necessary multiplication or division.

For example, in the problem, after distribution, the task of simplifying fractions like \( \frac{126}{7}h^{2} \) is essential to move towards a more refined form \( 18h^{2} \).

As you simplify expressions, aim to reduce unwanted complexity while maintaining the mathematical integrity of the initial expression.

Combining Like Terms

Combining like terms is a method to bring together terms that have the same variable raised to the same power. This process reduces expressions into a simpler form and is crucial for making equations easier to solve.

• Look for terms with the same variable and exponent, such as \(-15h\) and \(7h\).
• Add or subtract the coefficients of these like terms to simplify.

In the example provided, terms \(-15h\) and \(7h\) are combined by summing their coefficients: \(-15 + 7 = -8\), resulting in \(-8h\).
Ultimately, this leads to a concise expression \(18h^{2} - 8h\) which is easier to work with in further calculations.