Operations with algebraic fractions

Simplify the following algebraic fraction as far as possible by factorising the numerator and the denominator:

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

Aoife has to simplify the algebraic fraction $\frac{x^2 - 4x}{x^2 - 16}$ on an exam. Her working was as follows: "Since $x^2$ appears in both the numerator and the denominator, I cancel both terms and I am left with $\frac{-4x}{-16}$, which by simplifying the negative signs and dividing by $4$ results in $\frac{x}{4}$".

1. a) Explain in detail what conceptual error Aoife makes regarding the fundamental property of simplifying algebraic fractions (cancelling terms being added/subtracted versus cancelling factors).
2. b) Choose a specific numerical value for $x$ (for example, $x = 5$) that serves as a counterexample to prove that the original fraction and the result obtained by Aoife do not have the same value.
3. c) Carry out the correct simplification by first fully factoring the numerator and the denominator.

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 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:

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

Find a second-degree polynomial $P(x)$ that makes the following equality of operations with algebraic fractions true:

1. a) State the division as a multiplication by the reciprocal fraction and deduce the factored expression of $P(x)$ after simplifying common factors.
2. b) Write the polynomial $P(x)$ in its expanded polynomial form ($a \times x^2 + b \times x + c$).
3. c) Indicate with justification which real values of $x$ must be excluded from the domain so that none of the intermediate fractions or divisions are undefined (zero denominators).

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

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:

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

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:

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

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

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

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:

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

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.

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:

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

To simplify the following complex fraction:

two different algebraic strategies are proposed:

• Method 1: Work separately on the overall numerator and denominator, obtaining in each one a single irreducible fraction with its respective least common multiple, and carry out the division at the end by multiplying by the reciprocal fraction.

• Method 2: Simultaneously multiply the main numerator and denominator by the overall least common multiple of all the partial denominators present, eliminating all internal fractions in a single step.

1. a) Carry out the complete simplification applying both methods and verify that you arrive at exactly the same irreducible result.
2. b) Compare the number of steps, intermediate factorisations, and potential error points in handling signs for each method. Justify which of the two is more efficient in this case.

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 $\frac{v^2 - w^2}{v}$.

In an electrical circuit, two resistors $R_1$ and $R_2$ are connected in parallel. The equivalent resistance $R_{eq}$ satisfies the relation $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. If the values of the resistors are 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 $x$.

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 $C_m$.

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 centre 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 $f$.

In a solar photovoltaic self-consumption installation on a farm in County Tipperary, two branches of panels whose internal resistances are $R_1$ and $R_2$ are connected in parallel. The equivalent resistance $R_{eq}$ of the circuit is given by the physical relation:

Due to thermal fluctuations during the day, the resistances of each branch (in ohms) depend on the panel temperature $T$ (in degrees Celsius, with $T > 10$) according to the rational models:

1. a) Write the expressions for $\frac{1}{R_1}$ and $\frac{1}{R_2}$ by first factoring their terms, and calculate the sum $\frac{1}{R_{eq}}$ by reducing to the least common multiple of the denominators.
2. b) Obtain the simplified expression for $R_{eq}$ as a single algebraic fraction in terms of $T$.
3. c) If during the hours of maximum summer insolation a temperature of $T = 30\text{ }^\circ\text{C}$ is recorded, calculate the exact value of $R_{eq}$ expressed as an irreducible fraction and determine whether it remains below the regulatory safety limit of $0.7$ ohms.

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 $R_{eq}$.

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 $h$.