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Magazine Chapter 7: Problem 55
Perform the indicated operations. $$ \frac{x^{3}+7 x^{2}}{3 x^{3}-x^{2}} \div \frac{5 x^{2}+36 x+7}{9 x^{2}-1} $$
Short Answer
Expert verified
The simplified expression is \(\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\).
Step by step solution
01
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.
02
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 \(\frac{9 x^{2}-1}{5 x^{2}+36 x+7}\).
03
Multiply the Rational Expressions
The multiplication of two fractions involves multiplying the numerators together and the denominators together. Our new expression is: \[\frac{(x^3 + 7x^2) \cdot (9x^2 - 1)}{(3x^3 - x^2) \cdot (5x^2 + 36x + 7)}\]
04
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.
05
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 \(x^2(3x - 1)\):\[\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\]
06
Final Expression
The remaining expression is simplified to:\[\frac{(x + 7)(3x + 1)}{5x^2 + 36x + 7}\]No further simplification is possible.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(x^2(3x - 1)\).
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 \(\frac{9x^2 - 1}{5x^2 + 36x + 7}\).
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.
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鈥檚 how simplification works:
Here鈥檚 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.
