Exercises: Algebraic fractions

Simplify the following algebraic fraction using algebraic identities to factorise the numerator:

State for which value of $x$ the original expression has no mathematical meaning.

Factorise by taking out the common factor in both the numerator and denominator to fully simplify the following expression:

A very common mistake among students is "crossing out" terms that are being added. Look at the following diagram and explain, without doing complex calculations, why the given equality is false:

What condition should be met for the simplification to be correct?

Calculate the sum of the following algebraic fractions. Since they have the same denominator, remember that you only need to operate on the numerators:

Calculate the following product of algebraic fractions and simplify the result before multiplying out the terms if possible:

Solve the following division of algebraic fractions by transforming it into a cross product and simplifying the final result:

The area of a rectangular advertising billboard is given by the polynomial expression $A(x) = x^2 - 1$. If we know that the base of the billboard measures $x + 1$, find the simplified algebraic expression that represents the height $h$.

A courier company calculates the shipping cost for $x$ packages using the formula $C(x) = 2x^2 + 10x$ euros. Write the algebraic fraction that represents the average cost per package and simplify it as far as possible.

Note: The average cost is calculated by dividing the total cost $C(x)$ by the number of packages $x$.

In a Christmas exam, Cian simplifies the algebraic fraction $\frac{x^2 - 9}{x^2 - 3x}$ by writing the following on his paper:

1. a) Explain with algebraic rigor why Cian's first step is a serious error: why can't you cross out terms that are being added/subtracted in a fraction? To convince Cian, provide a numerical counterexample by calculating the value of the original fraction and that of his result for $x = 4$.
2. b) Carry out the correct simplification by first factoring the numerator and the denominator.
3. c) State for which values of $x$ the original fraction is not defined and check whether those values caused the same problem in Cian's erroneous expression.

Design an algebraic fraction $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials with real coefficients, that simultaneously satisfies the following four conditions:

1. Its denominator vanishes solely and exclusively for $x = 3$ and for $x = -1$.

2. The degree of both the numerator and the denominator is 2.

3. The fraction can be simplified by canceling the common factor $(x - 3)$.

4. After simplifying it completely, the resulting irreducible expression is $\frac{x + 2}{x + 1}$.

Write the original fraction $\frac{P(x)}{Q(x)}$ with the polynomials expanded (without indicated products) and justify step by step why it satisfies each of the four requested conditions.

A team of aerospace engineers is analysing the strength of a new composite material. The relationship between deformation and applied load is given by the following algebraic fraction:

Factorise the numerator and the denominator to fully simplify the expression and facilitate the team's subsequent calculations.

In a study on the spread of news on social media, it has been determined that users' response time (in seconds) depends on two variables combined in the following operation:

Perform the operation by putting the fractions over a common denominator and simplify the result to obtain a single algebraic fraction that models this behaviour.

An urban logistics company uses drones to deliver packages. The base area of the transport boxes is rectangular and its dimensions, in decimetres, are given by the expressions shown in the image. Calculate the simplified expression representing the total area of the box's base by multiplying both dimensions.

The dimensions are $Largo = \frac{x^2 - 1}{x}$ and $Ancho = \frac{x+2}{x-1}$. Calculate the area of the rectangle by simplifying the product of both fractions.

A financial analyst is comparing the profitability of two investment funds. The difference in profitability between them is expressed as the quotient of two algebraic fractions:

Work out the division and simplify the final result so that the analyst can present a clearer report to their clients.

In the design of a new smart park, architects want the ratio between the lawn area and the play area to follow a specific rule. For the project to be approved, the following algebraic fraction must be able to be simplified:

Calculate the value of the constant $k$ so that the numerator is divisible by the denominator and perform the simplification.

A group of Year 12 students is carrying out a chemistry experiment on reaction rates. They have obtained a complex expression that they need to simplify before entering the data into their simulation software:

Work out the expression following the order of operations and simplify the final result.

A microchip is being designed where the data flow through a channel is governed by the sum of two reciprocals of capacity. The resulting expression is:

Simplify this "fraction of fractions" (complex fraction) to obtain a single linear or quadratic algebraic expression defining the total flow.

Two cyclists cover the same distance $d$. The first travels at an average speed of $x$ km/h and the second at an average speed of $x+5$ km/h. The difference in time (in hours) it takes them to complete the journey is given by the expression:

If the distance is constant $d = 100$ km, subtract the algebraic fractions and obtain a single expression relating the time difference to the speed $x$.

To solve the following combined operation with algebraic fractions:

two 5th Year students propose different approaches:

• Aoife's method: First carries out the subtraction inside the parentheses by finding a common denominator and, then, performs the division by multiplying by the reciprocal fraction and simplifying.

• Seán's method: First inverts the second fraction to turn the division into a multiplication, applies the distributive property by multiplying that fraction by each of the terms inside the parentheses, and simplifies each product before subtracting.

1. a) Solve the operation following Aoife's procedure step by step.
2. b) Solve the operation following Seán's procedure step by step.
3. c) Compare both approaches: which of the two methods required dealing with polynomials of lower degree and minimizes the risk of making sign errors? Justify which one you would recommend to a classmate.

A small company in Athlone calculates the average unit assembly cost (in hundreds of euros) of a solar inverter model based on the number $x$ of batches produced (measured in hundreds of units, with $x > 1$), using the expression:

1. a) Estimate: Before working with pen and paper, analyze the behavior of each fraction if the factory reached a huge production (for example, imagining $x = 100$). Reasonably estimate what integer value the average unit cost $C(x)$ will tend to by mentally comparing the highest-degree terms of each fraction.
2. b) Calculate: Perform the subtraction algebraically by factoring the denominators, finding a common denominator, and simplifying until obtaining a single irreducible fraction.
3. c) Check: Substitute $x = 100$ into your simplified fraction to obtain the exact numerical value (you may round to two decimal places). Is it consistent with the estimation you made in part a)? What economic advantage does increasing the production volume offer according to this model?

Fully simplify the following algebraic expression, first factorising all polynomials using algebraic identities and Ruffini's rule:

Determine the values of the real parameters $A$ and $B$ so that the following equality between fractions is valid for any value of $x$ that does not make the denominators zero:

In the marketing department of a cosmetics company, they have determined that the total cost (in euros) of a social media advertising campaign is given by the expression $C(x) = \frac{2x^2 + 10x}{x - 1}$, where $x$ is the number of days the campaign lasts ($x > 1$). On the other hand, the number of new customers reached is estimated by $N(x) = \frac{x^2 + 5x}{x^2 - 1}$.

Obtain and simplify the expression for the "Investment per Customer" ($IC$), defined as the quotient between the total cost and the number of customers reached: $IC(x) = \frac{C(x)}{N(x)}$.

Perform the following combined operation and simplify the result to obtain an irreducible fraction:

Prove, by expanding and simplifying the left-hand side of the equality, that the following expression is an identity (the result is constant):

"Continued fractions" are expressions where the denominators themselves contain other fractions. Simplify the following structure to convert it into a single rational algebraic fraction:

A grain storage tank is to be designed in the shape of a rectangular prism. The relationship between the volume $V$ and the base area $B$ defines the height $h = \frac{V}{B}$.

If the volume of the tank is given by $V(x) = x^3 - 7x + 6$ and the base area by $B(x) = x^2 + x - 6$, express the height $h$ as a simplified polynomial. For what value of $x$ would the height of the tank be $5$ meters?

Find the value of the coefficients $a$ and $b$ knowing that the polynomial $P(x) = x^3 + ax^2 + bx + 12$ is divisible by $(x - 2)$ and that simplifying the algebraic fraction $\frac{P(x)}{x^2 - x - 6}$ yields a first-degree polynomial. Once $a$ and $b$ have been found, carry out this simplification.

A high-speed train covers the $360\text{ km}$ route between Dublin Heuston and Cork Kent. On the outbound trip, it travels at a constant average speed $v$ (in $\text{km/h}$). On the return trip, due to a Met Éireann warning for strong crosswind gusts in the Golden Vale region, the train is forced to reduce its average speed by $30\text{ km/h}$.

1. a) Write the algebraic fraction that represents the total round-trip time taken by the train as a function of $v$, and reduce it to a single irreducible fraction.
2. b) The overall average speed of the entire round trip is not calculated by adding speeds, but by dividing the total distance traveled ($720\text{ km}$) by the total time spent. Determine the irreducible algebraic expression of this overall average speed, $V_m(v)$.
3. c) A spokesperson for the company claims that the overall average speed is simply the arithmetic mean of both speeds: $\frac{v + (v - 30)}{2} = v - 15$. Check whether this is true by calculating the algebraic difference $(v - 15) - V_m(v)$ as a single fraction. Prove that for any admissible real speed ($v > 30\text{ km/h}$) this difference is strictly positive, and explain what this means in practice for timetable planning.

The logistics department of an express courier company models the waiting time (in minutes) at a distribution hub using the sum of two partial fractions:

where $x$ is the number of workers on shift ($x > 2$) and $A$ and $B$ are fixed real constants.

1. a) After finding a common denominator and adding both fractions, the technical team obtains the single expression:

Set up the corresponding algebraic identity and equate the coefficients of the numerator to deduce, with justification, the exact values of $A$ and $B$.

1. b) To optimize the system, it is proposed to multiply $f(x)$ by the adjustment factor $g(x) = \frac{x^2 - 4}{7x^2 + 22x + 3}$. Carry out the combined operation $f(x) \times g(x)$, factoring all polynomials beforehand to simplify the expression as much as possible.
2. c) Accurately determine the domains of definition: for which values of $x$ can the original product function $f(x) \times g(x)$ not be evaluated, whereas the numerical value could indeed be calculated in the simplified fraction obtained in part b)? Reason why simplifying an algebraic fraction can hide restrictions of the original domain.