Problem 55 - Algebra A Combined Function

Question

Perform the indicated operations. $$ \frac{x^{3}+7 x^{2}}{3 x^{3}-x^{2}} \div \frac{5 x^{2}+36 x+7}{9 x^{2}-1} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\).

Step 1: Interpret the Problem

The problem involves the division of two rational expressions: \(\frac{x^{3}+7 x^{2}}{3 x^{3}-x^{2}}\) and \(\frac{5 x^{2}+36 x+7}{9 x^{2}-1}\). To divide these expressions, we multiply the first fraction by the reciprocal of the second.

Step 2: Find the Reciprocal

The reciprocal of \(\frac{5 x^{2}+36 x+7}{9 x^{2}-1}\) is \(\frac{9 x^{2}-1}{5 x^{2}+36 x+7}\). This means we need to multiply \(\frac{x^{3}+7 x^{2}}{3 x^{3}-x^{2}}\) by \(\frac{9 x^{2}-1}{5 x^{2}+36 x+7}\).

Step 3: Multiply the Rational Expressions

The multiplication of two fractions involves multiplying the numerators together and the denominators together. Our new expression is: \[\frac{(x^3 + 7x^2) \cdot (9x^2 - 1)}{(3x^3 - x^2) \cdot (5x^2 + 36x + 7)}\]

Step 4: Simplify the Expression

Factor each part if possible to simplify the expression. First, factor the numerators and denominators: - The numerator \(x^3 + 7x^2\) can be factored as \(x^2(x + 7)\).- The numerator \(9x^2 - 1\) is a difference of squares, factored as \((3x - 1)(3x + 1)\).- The denominator \(3x^3 - x^2\) can be factored as \(x^2(3x - 1)\).- The denominator \(5x^2 + 36x + 7\) does not factor neatly, so we leave it as is.

Step 5: Cancel Common Factors

Substitute the factored forms into the expression:\[\frac{x^2(x + 7)(3x - 1)(3x + 1)}{x^2(3x - 1)(5x^2 + 36x + 7)}\]Cancel out the common factor \(x^2(3x - 1)\):\[\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\]

Step 6: Final Expression

The remaining expression is simplified to:\[\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\]No further simplification is possible.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when working with rational expressions. It involves breaking down a polynomial into simpler components, or factors, that when multiplied together give the original polynomial.

• For instance, consider the polynomial \(x^3 + 7x^2\). We notice that both terms have a common factor of \(x^2\), which means we can factor it out. This gives us \(x^2(x + 7)\).
• In the case of \(9x^2 - 1\), it is a difference of squares, which is factored as \((3x - 1)(3x + 1)\). Differences of squares take the form \(a^2 - b^2\) and can be factored as \((a - b)(a + b)\).
• Similarly, \(3x^3 - x^2\) can be factored by first taking out the common factor of \(x^2\), resulting in \(x^2(3x - 1)\).

Factoring simplifies the process of dividing rational expressions by making it easier to identify and eliminate common factors.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key to dividing rational expressions. The reciprocal is simply swapping the numerator and the denominator of the fraction.
When dividing by a fraction, you actually multiply by its reciprocal. For example, if the fraction is \(\frac{5x^2 + 36x + 7}{9x^2 - 1}\), its reciprocal is \(\frac{9x^2 - 1}{5x^2 + 36x + 7}\).

• This means instead of dividing by the original fraction, you multiply by the reciprocal.
• This swap turns the division operation into multiplication, which is generally easier to manage, especially when dealing with polynomials.

Switching from division to multiplication using the reciprocal makes calculations more straightforward and often leads to a simpler form of the result.

Simplifying Expressions

Simplifying expressions aims to reduce a mathematical expression to its simplest form. This involves several steps, including factoring and canceling common factors.
Here’s how simplification works:

• After factoring each part of a rational expression, as seen with expressions like \(x^2(x + 7)(3x - 1)(3x + 1)\) over \(x^2(3x - 1)(5x^2 + 36x + 7)\), it allows us to easily identify and cancel out common factors, such as \(x^2(3x - 1)\).
• This cancellation process reduces the expression, leaving fewer terms to work with, which is especially handy in complex equations or expressions.
• The simplified expression \(\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\) is easier to interpret and further modifications or calculations become more manageable.

The goal of simplifying expressions is to make them easier to understand and work with, eliminating any unnecessary complexity.