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Chapter 6: Problem 95

Add or subtract as indicated. $$\frac{r+s}{3 r^{2}+2 r s-s^{2}}-\frac{s-r}{6 r^{2}-5 r s+s^{2}}$$

Short Answer

Expert verified
\[6r^3 + r^2s^2 - 11rs^2]

Step by step solution

01

Find a common denominator

To add or subtract fractions, first find a common denominator. The denominators are \(3r^2 + 2rs - s^2\) and \(6r^2 - 5rs + s^2\).
02

Factor the denominators

Factor each denominator if possible. The polynomials do not factor easily, so the common denominator is the product: \[(3r^2 + 2rs - s^2)(6r^2 - 5rs + s^2)\]
03

Express each fraction with the common denominator

Rewrite each fraction with the common denominator: \[\frac{(r+s)(6r^2 - 5rs + s^2)}{(3r^2 + 2rs - s^2)(6r^2 - 5rs + s^2)} - \frac{(s-r)(3r^2 + 2rs - s^2)}{(3r^2 + 2rs - s^2)(6r^2 - 5rs + s^2)}\]
04

Combine the numerators

Subtract the numerators: \[(r+s)(6r^2 - 5rs + s^2) - (s-r)(3r^2 + 2rs - s^2)\]
05

Distribute and simplify

Distribute each term: \[6r^3 + r^2s^2 - 11rs^2\]Simplify if possible: \[6r^3 + r^2s^2 - 11rs^2\](3r^2 + 2rs - s^2)(6r^2 - 5rs + s^2)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
To add or subtract fractions, they must have the same denominator. This denominator is known as the 'common denominator'. In the given problem, the denominators are different: \(3r^2 + 2rs - s^2\) and \(6r^2 - 5rs + s^2\). To find a common denominator:
  • Ensure both denominators are the same.
  • If not, multiply each fraction by a form of one that will create the same denominator.
This process ensures that both fractions can be easily combined by aligning their bases.
Factoring Polynomials
Factoring polynomials is essential for simplifying and solving polynomial equations. Although the polynomials in our problem don鈥檛 factor easily, understanding how factoring works is beneficial:
  • Rewrite the polynomial as a product of simpler polynomials.
  • For example, \(3r^2 + 2rs - s^2\) can be checked for factors like \((ar + bs)(cr + ds)\).
Factoring helps in identifying the least common multiple, which in turn makes finding a common denominator less complex.
Subtracting Fractions
Once the fractions have the same denominator, we can subtract them:
  • Rewrite each fraction using the common denominator.
  • Subtract the numerator of the second fraction from the numerator of the first fraction.
  • The common denominator remains unchanged.
In our problem, we subtract the numerators as follows: \((r+s)(6r^2 - 5rs + s^2) - (s-r)(3r^2 + 2rs - s^2)\).
Distributing Terms
Distribution involves multiplying each term inside a parenthesis by the term outside it:
  • Apply the distributive property: \(a(b + c) = ab + ac\).
  • Carefully multiply each term in the parentheses to maintain accuracy.
In our step, we distribute as follows:
\((r+s)(6r^2 - 5rs + s^2) - (s-r)(3r^2 + 2rs - s^2)\) becomes:
\( r \times 6r^2 + r \times -5rs + r \times s^2 + s \times 6r^2 + s \times -5rs + s \times s^2 - (s \times 3r^2 + s \times 2rs - s \times s^2)\),
which simplifies further to combine like terms.

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Most popular questions from this chapter

A student was asked to simplify the rational expression $$ \frac{2 x+y}{4 x+2 y} $$ and wrote the incorrect answer \(\frac{x+y}{4 x+y}\) WHAT WENT WRONG? What is the correct answer?

Which rational expression is not equivalent to \(\frac{x-3}{4-x} ?\) A. \(\frac{3-x}{x-4}\) B. \(\frac{x+3}{4+x}\) C. \(-\frac{3-x}{4-x}\) D. \(-\frac{x-3}{x-4}\)

Write each rational expression in lowest terms. $$ \frac{7 x-21}{63-21 x} $$

Add or subtract as indicated. $$\frac{8}{(3 r-1)^{2}}+\frac{2}{3 r-1}-6$$

Multiply or divide as indicated. $$ \frac{a^{3}-8 b^{3}}{a^{2}-a b-6 b^{2}} \cdot \frac{a^{2}+a b-12 b^{2}}{a^{2}+2 a b-8 b^{2}} $$
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