Problem 41 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{2 a^{2}}{3 a^{3}+24 a^{2}+45 a}-\frac{10 a+48}{3 a^{3}+24 a^{2}+45 a} $$

Step-by-Step Solution

Answer

Simplified form: \( \frac{2(a^2 - 5a - 24)}{3a(a^2 + 8a + 15)} \).

Step 1: Identify the common denominator

Both fractions have a common denominator of \(3a^3 + 24a^2 + 45a\).

Step 2: Combine the fractions

Rewrite the expression by combining the two fractions under the common denominator. \[ \frac{2a^2 - (10a + 48)}{3a^3 + 24a^2 + 45a} \]

Step 3: Simplify the numerator

Distribute the negative sign and combine like terms in the numerator.\[ 2a^2 - 10a - 48 \]

Step 4: Factor the numerator and the denominator

Factor the numerator, if possible, and factor the denominator. The numerator \(2a^2 - 10a - 48\) can be factored as \(2(a^2 - 5a - 24)\). The denominator \(3a^3 + 24a^2 + 45a\) can be factored by taking out a common factor of \(3a\), giving: \[ 3a(a^2 + 8a + 15) \].

Step 5: Simplify the fraction

Cancel any common factors in the numerator and the denominator, if possible. In this case, there are no common factors to cancel, hence:\[ \frac{2(a^2 - 5a - 24)}{3a(a^2 + 8a + 15)} \]

Key Concepts

Common Denominator

When working with fractions, a common denominator is essential for combining them. This is because the denominators must match to allow proper addition or subtraction of fractions. Think of the denominator as a 'common ground' for the fractions.
In our exercise, the denominator of both fractions is already the same: \(3a^3 + 24a^2 + 45a\).
This simplifies our task greatly, as we can directly move on to the next step of combining the fractions.

Combining Fractions

Once you have a common denominator, you can combine the fractions by simply adding or subtracting their numerators while keeping the common denominator unchanged. This looks like:

• \( \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} \) for addition
• \( \frac{a}{b} - \frac{c}{b} = \frac{a-c}{b} \) for subtraction

In our example, we need to subtract: \[ \frac{2a^2}{3a^3 + 24a^2 + 45a} - \frac{10a + 48}{3a^3 + 24a^2 + 45a} = \frac{ 2a^2 - (10a + 48) }{3a^3 + 24a^2 + 45a} \] Remember to distribute the negative sign in the numerator as shown.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into products of simpler polynomials. This is an essential step for simplifying expressions.
In our solution, we need to factor both the numerator and the denominator: \[2a^2 - 10a - 48 \] and \[3a^3 + 24a^2 + 45a.\]
For the numerator, factor out the greatest common factor first if possible: \[2(a^2 - 5a - 24)\]
For the denominator: \[3a(a^2 + 8a + 15) \]
This makes the fraction easier to work with and simplifies further computations.

Numerator and Denominator Simplification

After factoring, look for any common factors between the numerator and the denominator. Simplification involves canceling out these common factors to reduce the fraction to its simplest form.
In some cases, as with our example: \[ \frac{2(a^2 - 5a - 24)}{3a(a^2 + 8a + 15)} \] There are no common factors that can be canceled. If there were any, you would simply reduce the fraction by dividing both the numerator and the denominator by the common factor. Simplifying fractions makes them easier to interpret and use in further calculations.