← Algebra II

Unit 5: Rational Expressions and Equations

Fractions built out of polynomials, from simplifying and combining them to solving with them and catching the fakes.

8 topics · 31 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Simplifying Rational Expressions

  1. How it worksHow does factoring both the top and bottom of a fraction reveal which pieces actually cancel?
  2. Worked exampleFind the simplified form of (x2 - 9) over (x2 + x - 6).
  3. Common mistakeWhy can't you cancel the x2 in (x2 + 3) over (x2 + 5) even though it appears in both places?
  4. Going furtherWhat value of x has to be excluded from the simplified expression that wasn't obvious before factoring?
Your turn1 of 2
Find the simplified form of (x2 - 72) over (x2 + 7x - 7x - 7·7).

Multiplying and Dividing Rational Expressions

  1. How it worksHow do you factor everything first before multiplying two rational expressions together?
  2. Worked exampleFind the product of (x2 - 4) over (x + 1) and (x + 1) over (3x - 6).
  3. Worked exampleFind the quotient of (x2 - 25) over (x + 3) divided by (x - 5) over (2x + 6).
Your turn1 of 2
Find the product of (x2 - 72) over (x + 6) and (x + 6) over (9(x - 7)).

Adding and Subtracting Rational Expressions with Unlike Denominators

  1. How it worksHow do you build a least common denominator out of two different polynomial denominators?
  2. Worked exampleFind the sum of 3 over (x - 2) and 5 over (x + 1).
  3. Common mistakeWhy can't you just add the numerators and add the denominators straight across?
  4. Going furtherWhat happens to the least common denominator when one denominator's factors are already inside the other?
Your turn1 of 2
Find the sum of 7 over (x - 7) and 8 over (x + 4).

Simplifying Complex Fractions

  1. How it worksHow does multiplying every term by the overall least common denominator clear a complex fraction in one move?
  2. Worked exampleFind the simplified form of the complex fraction with numerator (1/x + 1) and denominator (1/x - 1).
  3. Common mistakeWhy can't you simplify a complex fraction by just canceling something from the top with something from the bottom?
  4. Ties togetherHow does clearing a complex fraction with its overall LCD use the same LCD idea as solving a rational equation?
Your turn1 of 2
Find the simplified form of the complex fraction with numerator (1/x + 7) and denominator (1/x - 7).

Solving Rational Equations

  1. How it worksHow do you multiply every term of a rational equation by the least common denominator to clear the fractions?
  2. Worked exampleSolve (3 over x) + (2 over x + 1) = 1 for x.
  3. Common mistakeWhy doesn't every value you get from clearing denominators actually solve the original equation?
Your turn1 of 2
Solve 4 over (x - 8) = 8 over (x - 3) for x.

Extraneous Solutions in Rational Equations

  1. How it worksHow does multiplying both sides by an expression that could equal zero smuggle a fake solution into the answer?
  2. Worked exampleSolve x over (x - 2) = 2 over (x - 2) and show why the resulting solution has to be thrown out.
  3. Common mistakeWhy isn't checking a rational equation's solution back in the original equation an optional step?
  4. Common mistakeWhy doesn't getting a clean, whole-number answer guarantee that a rational equation's solution is actually valid?
Your turn1 of 2
Clear the denominators in x over (x - 7) = 7 over (x - 7) and solve for x. (Don't worry yet about whether the solution is valid.)

Graphing Rational Functions and Their Asymptotes

  1. How it worksHow do the zeros of a rational function's denominator become vertical asymptotes on its graph?
  2. Worked exampleFind the vertical and horizontal asymptotes of f(x) = (2x + 1) over (x - 3).
  3. Common mistakeWhy doesn't every zero of the denominator turn into a vertical asymptote?
  4. Going furtherWhat happens to a rational function's graph at an x-value where both the numerator and denominator hit zero together?
  5. Ties togetherHow does a rational function's end behavior connect to polynomial long division when the numerator's degree is larger?
Your turn1 of 2
Rewrite f(x) = (7x + 6)/(x - 8) in the form A + B/(x-c), by polynomial division. Write f(x) in this form, as a function of x.

Direct and Inverse Variation

  1. How it worksHow does the constant of variation turn a verbal description into an equation like y = k over x?
  2. Worked exampleFind the constant of variation and write the equation if y varies inversely with x, and y = 4 when x = 15.
  3. Common mistakeWhy isn't inverse variation just direct variation with a negative constant?
  4. Ties togetherHow does the inverse variation equation y equals k over x connect to the rational function graphs studied earlier?
Your turn1 of 2
y varies inversely with x. If y = 10 when x = 11, write y as a function of x.