Operations with Algebraic Fractions

Simplify the following algebraic fraction as far as possible by factorising 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:

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

Divide the following algebraic fractions and simplify the result obtained:

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

Add the following algebraic fractions with common denominators:

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

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

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

Add the following algebraic fractions with linear denominators:

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

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

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

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

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.

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

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

Carry out the following combined operation of algebraic fractions and simplify as far as possible:

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

Perform the following combined operations respecting the order of operations and simplify the result as far as possible:

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

Show that the following algebraic equality holds for any values of the variables $x$ and $y$ that do not make the denominators zero, by carrying out the operations and simplifying the left-hand side:

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

Carry out the indicated operations involving three variables and prove that the final result simplifies to a constant integer:

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

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

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

Simplify the following continued fraction expression by working from the bottom up until an irreducible fraction is obtained:

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

A motorboat travels along a canal from a jetty to a buoy at a speed $v+w$ (where $w$ is the downstream current speed) and returns at a speed $v-w$ (upstream). The distance between the points is $d$. Obtain the simplified expression for the average speed of the round trip, given by $v_m = \frac{2d}{\frac{d}{v+w} + \frac{d}{v-w}}$, and check whether it is equivalent to the expression $\frac{v^2 - w^2}{v}$.

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

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 $C$ such 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 minimise the use of plastic material is given by the expression $S(x) = 2\pi x^2 + \frac{1000}{x}$. To analyse the rate of change of 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 $h$.