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 $C(x) = 40x^2 + 200x$ pesos. Write the algebraic fraction that represents the average cost per package and simplify it as much as possible.

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

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 behavior.

An urban logistics company uses drones to deliver packages. The base area of the transport boxes is rectangular and its dimensions, in decimeters, 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 $x$.

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 pesos) 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:

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

If the volume of the tank is given by $V(x) = x^3 - 7x + 6$ and the area of the base by $B(x) = x^2 + x - 2$, 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.