Algebraic fractions

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

$$\frac{x^2 - 25}{x - 5}$$

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:

$$\frac{4x^2 + 8x}{2x^2 - 6x}$$

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:

$$\frac{x^2 - 9}{x^2 - 3x} = \frac{\cancel{x^2} - 9}{\cancel{x^2} - 3x} = \frac{-9}{-3x} = \frac{3}{x}$$

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.

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

$$\frac{x + 3}{5} \times \frac{10}{x^2 - 9}$$

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

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

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.

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

$$\frac{3x + 1}{x + 2} + \frac{x + 7}{x + 2}$$

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:

$$\frac{3x}{x-2} - \frac{x-1}{x+2} - \frac{8}{x^2-4}$$

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.

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:

$$T(x) = \frac{d}{x} - \frac{d}{x+5}$$

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

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:

$$\frac{8x - 11}{x^2 - x - 6} = \frac{A}{x - 3} + \frac{B}{x + 2}$$

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:

$$F(x) = \frac{x^2 + kx - 12}{x - 3}$$

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

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.

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

$$\frac{x^4 - 13x^2 + 36}{x^3 - x^2 - 9x + 9} \times \frac{x^2 - 6x + 9}{x^2 + x - 6}$$