Exercises: Algebraic fractions

Simplify the following algebraic fraction by factoring out the common factor in the numerator and denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$

Check, by cross-multiplying, if the algebraic fractions $\frac{x-3}{x+1}$ and $\frac{x^2-9}{x^2+4x+3}$ are equivalent.

Add the following algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$

Multiply the following algebraic fractions and write the result in simplest form: $\frac{x}{x+2} \cdot \frac{x^2-4}{x^2}$

Divide the following algebraic fractions and simplify: $\frac{2x+2}{x-1} : \frac{4x+4}{x^2-1}$

Evaluate the algebraic fraction $\frac{x^2 - 3x + 2}{x^2 + 1}$ for $x = -2$.

Simplify the following algebraic fraction by applying algebraic identities: $\frac{x^2 - 25}{x^2 - 10x + 25}$

Perform the following operation and simplify the result: $\frac{x^2+5x}{x+3} - \frac{x^2-9}{x+3}$

Simplify the following algebraic fraction by first factoring the numerator and the denominator: $\frac{x^3 - 4x^2 + 4x}{x^3 - 2x^2 - 4x + 8}$

Perform the following operations by finding a common denominator and simplify the result: $\frac{2}{x-1} + \frac{x}{x+1} - \frac{2x}{x^2-1}$

The base of a rectangular solar panel measures $x+5$ feet and its height measures $x-5$ feet. The side length of a square solar panel measures $x$ feet. Write the algebraic fraction that represents the ratio of the area of the rectangular panel to the area of the square panel. Find the value of this fraction if $x = 10$ feet.

A hot water faucet takes $x$ hours to fill a bathtub, while the cold water faucet takes $x+3$ hours. Write a simplified algebraic fraction that represents the portion of the bathtub filled in one hour if both faucets are turned on at the same time.

Perform the following combined operation with algebraic fractions and simplify the result: $\left( \frac{1}{x} - \frac{2}{x+2} \right) : \frac{x-2}{x^2+2x}$

The total cost in dollars of manufacturing $x$ bicycles is given by the polynomial $C(x) = 3x^2 + 150x$. The total revenue in dollars from selling these $x$ bicycles is expressed by the polynomial $I(x) = 5x^2 + 200x$. Write the simplified algebraic fraction that represents the ratio of the revenue to the costs.

Subtract the following algebraic fractions and simplify fully: $\frac{x^2+x}{x^2-1} - \frac{x}{x-1}$

A train travels a distance of $120$ miles at a constant speed of $v$ mph. On the return trip, due to rain, its speed decreases by $10$ mph. Write the algebraic fraction that represents the total time of the round trip in terms of $v$, combining it into a single fraction.

Simplify the following expression fully: $\frac{x^2-1}{x^2+2x+1} \cdot \left( \frac{2x}{x-1} - \frac{x+1}{x} \right)$

Determine the value of the parameter $k$ so that the algebraic fraction $\frac{x^2 - kx + 10}{x-5}$ can be simplified (that is, the numerator is divisible by the denominator). Once $k$ is found, write the simplified expression.

Find the polynomial $P(x)$ that makes the following equality between algebraic fractions true: $\frac{x^2-9}{P(x)} = \frac{x-3}{x^2+2x}$

An open box is to be made from a rectangular sheet of cardboard with dimensions $x+6$ and $x+4$ inches, by cutting out squares with side length $2$ in. from the four corners and folding up the flaps. Write the simplified algebraic fraction that represents the ratio of the volume of the box to the area of its base.

Show, by expanding and simplifying the algebraic fractions, that the following expression is equal to $2ab$: $\frac{a^3-b^3}{a-b} - \frac{a^3+b^3}{a+b}$

In a chemistry laboratory, $x$ gallons of solution A, which costs $\$\frac{20}{x}$ per gallon, are mixed with $x+4$ gallons of solution B, which costs $\$\frac{30}{x+4}$ per gallon. Write the algebraic fraction that represents the price per gallon of the resulting solution, simplifying the expression fully.

Simplify the following algebraic expression in the form of a complex fraction: $1 + \frac{1}{1 + \frac{1}{x-1}}$

An athlete runs on a circular track. During the first half of the distance, they run at a speed of $v$ ft/s, and during the second half, they run at a speed of $v-2$ ft/s. Knowing that the average speed for the whole trip is calculated as total distance divided by total time, show using algebraic fractions that the average speed is $\frac{2v(v-2)}{2v-2}$ ft/s.