Problem 76 - Elementary Algebra

Question

\(-\frac{5}{6} b\left(12 b+\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

-10 b^2 - \frac{5}{24} b

Step 1: Distribute the Terms

First, distribute \(-\frac{5}{6} b\) to the terms inside the parenthesis. This means multiplying \(-\frac{5}{6} b\) with \(12 b\) and \(-\frac{5}{6} b\) with \(\frac{1}{4}\).

Step 2: Multiply \-\frac{5}{6} b\ with \12 b\

Multiply the coefficients \-\frac{5}{6}\ and \12\, and then multiply the \b\ terms: \[-\frac{5}{6} b \cdot 12 b = -\frac{5 \cdot 12}{6} b^2 = -10 b^2\].

Step 3: Multiply \-\frac{5}{6} b\ with \frac{1}{4}\

Multiply the coefficients \-\frac{5}{6}\ and \frac{1}{4}\, and then retain the \b\ term: \[-\frac{5}{6} b \cdot \frac{1}{4} = -\frac{5 \cdot 1}{6 \cdot 4} b = -\frac{5}{24} b\].

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to get the final expression: \[-10 b^2 - \frac{5}{24} b\].

Key Concepts

Multiplying Fractions

When you multiply fractions, you multiply the numerators together and the denominators together. For example, to multiply \(-\frac{5}{6}\) and \(\frac{1}{4}\), you do the following:

1. Multiply the numerators: \( -5 \cdot 1 = -5 \)
2. Multiply the denominators: \( 6 \cdot 4 = 24 \)
As a result, the product of \(-\frac{5}{6}\) and \(\frac{1}{4}\) is \(-\frac{5}{24}\).

Remember to always simplify the fractions if possible. However, in this case, \(-\frac{5}{24}\) is already in its simplest form.

Combining Like Terms

Like terms are terms that have the same variable raised to the same power. You can combine them by adding or subtracting their coefficients.

For example:
If you have \(-10 b^2\) and \(-\frac{5}{24} b\), you cannot combine them directly because they are not like terms.

Here's why:

• \(-10 b^2\) has the variable \(b\) raised to the power of 2.
• \(-\frac{5}{24} b\) has the variable \(b\) raised to the power of 1.

To combine like terms, ensure that the variables and their exponent (power) match. Only then can you add or subtract their coefficients.

Polynomial Expressions

A polynomial expression is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients.

For instance, in the exercise:
The expression: \( - \frac{5}{6} b (12 b + \frac{1}{4}) \) becomes a polynomial after distributing and combining the terms. Here's how:

• Distribute \(-\frac{5}{6} b\) over the terms inside the parentheses: \(-\frac{5}{6}b (12b) + (-\frac{5}{6}b) (\frac{1}{4})\).
• This gives you: \(-10 b^2\) and \(-\frac{5}{24} b\).
• The resulting expression: \(-10 b^2 - \frac{5}{24} b\) is a simplified polynomial.

Polynomials are often written in standard form, where the terms are arranged in descending order of their exponents. In this example, \(-10 b^2\) comes before \(-\frac{5}{24} b\), aligning with the standard form.