Exercises: 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:

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

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:

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

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.

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).

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 $x$.

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.

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

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

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 $f$.

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:

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

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 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.

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.

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.

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:

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

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

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

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

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 $R_{eq}$.

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

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

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

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

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 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 $h$.

Two pipelines from the Uisce Éireann distribution network discharge water simultaneously into a retention basin in Dublin. The modern pipeline $A$ takes $x$ hours to fill the entire basin on its own, while the older pipeline $B$, with a lower flow rate, takes $x + 4$ hours (with $x > 0$).

1. a) Estimate: Without performing exact algebraic operations, consider the case where $x = 6$ hours (pipe $A$ takes 6 hours and pipe $B$ takes 10 hours). Reason qualitatively between which two whole numbers of hours the total joint filling time $T$ must necessarily lie. (Hint: Compare the system's flow rate with two hypothetical situations: two pipelines identical to the fast one or two identical to the slow one).
2. b) Calculate: Knowing that hourly flow rates add up (satisfying the relation $\frac{1}{T(x)} = \frac{1}{x} + \frac{1}{x+4}$), operate and simplify to express the joint time $T(x)$ as a single irreducible algebraic fraction.
3. c) Check and generalise: Calculate the exact value of $T(6)$ from the formula found, convert it into hours and minutes, and check whether your estimate from section (a) was correct. Next, analyse the sign of the algebraic difference $T(x) - \frac{x}{2}$ and formally prove why the joint time can never drop below half the time of the faster pipe, whatever the value of $x$.

In an applied optics laboratory at the CSIC, a test bench containing a thin converging lens of known and fixed focal length $f$ ($f > 0$) is adjusted. The distance from the luminous object to the lens is $s$ (with $s > f$) and the distance from the lens to the focused image on a screen is $s'$. These variables satisfy the Descartes equation:

1. a) Solve for $s'$ as a single algebraic fraction in terms of $s$ and $f$. From it, obtain the simplified expression for the total distance between the object and the projection screen, $L(s) = s + s'$, as a single algebraic fraction with the denominator factored.
2. b) Inverse problem: To verify purely algebraically that the total separation between object and screen can never be less than four times the focal length ($L(s) \ge 4f$), set up the subtraction $D(s) = L(s) - 4f$, reduce to a common denominator, and fully factor the resulting numerator by recognising a notable identity.
3. c) Analyse the sign of the numerator and denominator of the fraction obtained for $D(s)$, taking into account that $s > f$. Conclude rigorously why $L(s) \ge 4f$ and deduce with justification what position the object must occupy ($s$ in terms of $f$) to achieve the distance between object and screen being exactly the minimum possible ($4f$).

Topic Challenge. Optimization of a domestic solar microgrid.

1. Data and phasors: Access the EirGrid Smart Grid Dashboard website and record the hourly solar photovoltaic generation in Ireland (or in your county, if data is available) at two key moments, associating them with power phasors in polar form: $z_1 = r_{1,\alpha}$ and $z_2 = r_{2,\beta}$. Calculate their quotient in polar form to evaluate the generation phase shift.

2. Complex algebra and trigonometry: Convert this quotient to rectangular (algebraic) form to multiply it by the grid impedance $Z = 2 + 1.5i$. Check the real and imaginary parts analytically by applying the angle difference trigonometric identities: $\cos(\alpha - \beta)$ and $\sin(\alpha - \beta)$.

3. Polynomial modeling: In GeoGebra, fit a cubic polynomial $P(t)$ to the daily power curve. Knowing the integer hour when solar production starts as a root, factor $P(t)$ by applying Ruffini's rule until it is completely factored. Then, calculate the derivative of the polynomial $P'(t)$ to find the exact hour of maximum production ($P'(t) = 0$).

4. Technical proposal: Write a short report with the GeoGebra graph and the calculations performed, justifying the optimal hour of consumption.

Assessment criteria:

• Precision in operations with complex numbers in polar and rectangular form, and in the angle difference expansion.

• Skill in Ruffini's rule, polynomial factorization, and calculation of the polynomial derivative.

• Rigour in using real data from EirGrid and clarity of argumentation in the final report.