Operations with Rational Expressions

Simplify the following algebraic fraction completely by factoring the numerator and the denominator:

$\frac{x^2 - 9}{x^2 + 6x + 9}$

Perform the following multiplication of algebraic fractions and express the result in its simplest form:

$\frac{2x}{x-1} \cdot \frac{x^2 - 1}{4x^2}$

Divide the following algebraic fractions and simplify the result obtained:

$\frac{x^2 + 2x}{x - 3} : \frac{x^2 - 4}{x^2 - 9}$

Prove that the following algebraic equality is true for any value of the variables $x$ and $y$ that does not make the denominators zero, by expanding the operations and simplifying the left-hand side:

$\left( \frac{x^2 - y^2}{x+y} \right) \cdot \left( \frac{x^3 + y^3}{x^2 - xy + y^2} \right) = x^2 - y^2$

Add the following algebraic fractions that have the same denominator:

$\frac{3x - 1}{x^2 + 1} + \frac{x + 5}{x^2 + 1}$

Subtract the following algebraic fractions by first finding a common denominator:

$\frac{5}{x+2} - \frac{3}{x^2 - 4}$

Perform the following addition of algebraic fractions with first-degree denominators:

$\frac{x}{x-2} + \frac{2}{x+3}$

Perform the addition and subtraction of the following algebraic fractions. Use Ruffini's rule and standard identities to first factor the denominators and find the least common multiple:

$\frac{2x}{x^3 - 3x^2 + 3x - 1} - \frac{1}{x^2 - 2x + 1} + \frac{3}{x-1}$

The average production cost of an electronic component is given by the expression $C_m(x) = \frac{x^3 + 2x^2 - 5x - 6}{x^2 - 4}$, where $x$ is the production volume in thousands of units. Due to a subsidy adjustment, a fee represented by the fraction $\frac{x^2-1}{x+2}$ must be subtracted. Calculate the new expression for the average cost by first simplifying the first fraction.

Perform the indicated operations with three variables and show that the final result simplifies to a constant integer:

$\frac{x^2}{(x-y)(x-z)} + \frac{y^2}{(y-x)(y-z)} + \frac{z^2}{(z-x)(z-y)}$

Work out the expression inside the brackets and then multiply to simplify the final result:

$\frac{x^2 - x}{x+1} \cdot \left( \frac{1}{x} + \frac{1}{x-1} \right)$

Perform the following combined operation of algebraic fractions and simplify completely:

$\frac{x^2+x}{x^2-1} - \frac{1}{x-1}$

Perform the following combined operations respecting the order of operations and simplify the result completely:

$\left( \frac{x+1}{x-1} - \frac{x-1}{x+1} \right) : \frac{4x}{x^2 - 1}$

Simplify the following complex algebraic fraction by first reducing the numerator and the denominator independently:

$\frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x}}$

Simplify the following iterated algebraic expression (continued fraction) by working from the deepest level upward, until obtaining an irreducible fraction:

$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x-1}}}$

A delivery drone travels from a warehouse to a residential area at a speed of $v+w$ (where $w$ is the tailwind speed) and returns at a speed of $v-w$ (headwind). The distance between the points is $d$. Obtain the simplified expression for the average speed of the entire journey, given by $v_m = \frac{2d}{\frac{d}{v+w} + \frac{d}{v-w}}$, and check if it is equivalent to the expression $\frac{v^2 - w^2}{v}$.

In an electrical circuit, two resistors $R_1$ and $R_2$ are connected in parallel. The equivalent resistance $R_{eq}$ satisfies the relationship $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. If the value of the resistors is given by the polynomials $R_1 = x^2 - 4$ and $R_2 = x^2 + 4x + 4$, obtain the simplified expression for the equivalent resistance $R_{eq}$ in terms of $x$.

The thin lens formula in optics states that $\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}$. If the distance from the object to the optical center is $s_o = \frac{x^2+x}{2}$ and the image distance is $s_i = \frac{x^2-1}{3}$, determine the simplified algebraic expression for the focal length $f$.

Two industrial 3D printers manufacture a batch of parts working as a team. The first printer takes $x$ hours to complete the job alone, and the second takes $x+3$ hours. The portion of the work they do together in one hour is the sum of their individual work rates. Write and simplify the algebraic fraction that represents this portion of work per hour.

The base of a rectangular solar panel is given by the expression $b = \frac{x^3 - x}{x^2 + 2x + 1}$ feet, and its height by $h = \frac{x^2 + x}{x^2 - 2x + 1}$ feet. Calculate the area of the solar panel, expressing the result as a single simplified algebraic fraction.

A mixed circuit has an equivalent resistance between terminals A and B given by the formula $R_{eq} = R_1 + \frac{1}{\frac{1}{R_2} + \frac{1}{R_3}}$. If the values of the resistors in ohms are given by the algebraic expressions $R_1 = \frac{x}{x+1}$, $R_2 = \frac{x^2-1}{x}$ and $R_3 = \frac{x^2-1}{2}$, calculate the simplified expression for the equivalent resistance $R_{eq}$.

In a chemistry laboratory, there are two solutions of the same acid. The first has a concentration of $C_1 = \frac{x}{x^2+2x+1}$ and the second $C_2 = \frac{2}{x^2-1}$. Equal volumes of both solutions are mixed, so the concentration of the resulting mixture is the arithmetic mean of both concentrations, that is, $C_m = \frac{1}{2}(C_1 + C_2)$. Obtain the simplified algebraic expression for the final concentration of the mixture $C_m$.

Determine the numerical values of the parameters $A$, $B$ and $C$ so that the following partial fraction decomposition identity holds for all $x$:

$\frac{5x^2 - x + 2}{x^3 - x^2 + x - 1} = \frac{A}{x-1} + \frac{Bx+C}{x^2+1}$

The surface area of a cylindrical container designed to minimize the use of plastic material is given by the expression $S(x) = 2\pi x^2 + \frac{1000}{x}$. To analyze the rate of change of the material with respect to the radius $x$, an engineer needs to calculate and simplify the expression for the difference quotient $\frac{S(x+h) - S(x)}{h}$. Perform this operation and express the result as a single simplified algebraic fraction in terms of $x$ and $h$.