Algebraic Fractions
A shipping store charges $C(x) = 40x^2 + 200x$ pesos to ship $x$ packages. How much does each package cost on average? The answer is a fraction with letters, and to handle it you do the same thing as with $\dfrac{10}{15}$: factor, cancel, add with a common denominator. Everything you already know about factoring is used here in a single block of calculation. At the end we return to the question of the cost per package.
Simplifying by factoring
Simplifying an algebraic fraction means canceling the factors that the numerator and denominator have in common. That is why the first step, always, is to factor both: common factor, difference of squares, perfect square trinomials, or trinomials of the form $x^2+bx+c$.
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…
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: $$\frac{4x^2 + 8x}{2x^2 - 6x}$$
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 $C(x) = 40x^2 + 200x$ pesos. Write the algebraic fraction that represents the average…
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 $x$.
Why you can't cancel addends
When an equality between fractions is false, a numerical value shows it quickly. Try $\dfrac{x+2}{2}$ and $x+1$: with $x=4$ you get $3$ versus $5$. A simplification is valid only when the number being canceled multiplies the entire numerator and the entire denominator.
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…
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?
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 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 $h$.
Multiplication and division
Just as with numerical fractions: multiplying is numerator times numerator and denominator times denominator; dividing is multiplying by the reciprocal fraction (or cross-multiplying). If you factor first, you can cancel as you go and avoid huge products.
Calculate the following product of algebraic fractions and simplify the result before multiplying out the terms if possible: $$\frac{x + 3}{5} \times \frac{10}{x^2 - 9}$$
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: $$\frac{x^2}{x - 1} : \frac{x}{x - 1}$$
Solve the following division of algebraic fractions by transforming it into a cross product and simplifying the final result:
$$\frac{x^2}{x - 1} : \frac{x}{x - 1}$$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…
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:
$$R(x) = \frac{x^3 - 7x^2 + 10x}{x^2 - 25}$$Factorise the numerator and the denominator to fully simplify the expression and facilitate the team's subsequent calculations.
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…
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…
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:
$$\left( \frac{x^2 - 9}{x^2 - 4x + 4} \right) : \left( \frac{x^2 + 3x}{x - 2} \right)$$Work out the division and simplify the final result so that the analyst can present a clearer report to their clients.
Addition and subtraction: least common multiple
With the same denominator you add the numerators and the denominator stays the same. With different denominators you need a common denominator: factor each one and take the LCM (each factor with its highest exponent).
Calculate the sum of the following algebraic fractions. Since they have the same denominator, remember that you only need to operate on the numerators: $$\frac{3x + 1}{x…
Calculate the sum of the following algebraic fractions. Since they have the same denominator, remember that you only need to operate on the numerators:
$$\frac{3x + 1}{x + 2} + \frac{x + 7}{x + 2}$$In a study on the spread of news on social media, it has been determined that users' response time (in seconds) depends on two variables combined in the following…
In a study on the spread of news on social media, it has been determined that users' response time (in seconds) depends on two variables combined in the following operation:
$$\frac{3x}{x-2} - \frac{x-1}{x+2} - \frac{8}{x^2-4}$$Perform the operation by putting the fractions over a common denominator and simplify the result to obtain a single algebraic fraction that models this behavior.
Two cyclists cover the same distance $d$. The first travels at an average speed of $x$ km/h and the second at an average speed of $x+5$ km/h. The difference in time (in…
Two cyclists cover the same distance $d$. The first travels at an average speed of $x$ km/h and the second at an average speed of $x+5$ km/h. The difference in time (in hours) it takes them to complete the journey is given by the expression:
$$T(x) = \frac{d}{x} - \frac{d}{x+5}$$If the distance is constant $d = 100$ km, subtract the algebraic fractions and obtain a single expression relating the time difference to the speed $x$.
Combined operations and complex fractions
When sums, products, and parentheses are mixed, the order of operations is respected: parentheses first, then multiplications and divisions, then additions and subtractions.
A group of Year 12 students is carrying out a chemistry experiment on reaction rates. They have obtained a complex expression that they need to simplify before entering…
A group of Year 12 students is carrying out a chemistry experiment on reaction rates. They have obtained a complex expression that they need to simplify before entering the data into their simulation software:
$$\frac{1}{x} + \frac{1}{x^2} \times \frac{x^2 - x}{x + 1}$$Work out the expression following the order of operations and simplify the final result.
Perform the following combined operation and simplify the result to obtain an irreducible fraction: $$\left( \frac{1}{x - 2} - \frac{x}{x^2 - 4} \right) : \left(…
Perform the following combined operation and simplify the result to obtain an irreducible fraction:
$$\left( \frac{1}{x - 2} - \frac{x}{x^2 - 4} \right) : \left( \frac{x}{x + 2} - \frac{2}{x^2 + 4x + 4} \right)$$Extension: complex fractions
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: $$C_{total} =…
A microchip is being designed where the data flow through a channel is governed by the sum of two reciprocals of capacity. The resulting expression is:
$$C_{total} = \frac{1}{\frac{1}{x} + \frac{1}{x+1}}$$Simplify this "fraction of fractions" (complex fraction) to obtain a single linear or quadratic algebraic expression defining the total flow.
"Continued fractions" are expressions where the denominators themselves contain other fractions. Simplify the following structure to convert it into a single rational…
"Continued fractions" are expressions where the denominators themselves contain other fractions. Simplify the following structure to convert it into a single rational algebraic fraction:
$$x - \frac{1}{1 - \frac{1}{1 - \frac{1}{x}}}$$Extension: factoring with synthetic division and parameters
If the polynomial has degree 3 or higher, look for an integer root (a divisor of the constant term), divide using synthetic division, and factor what is left. With $P(a)=0$ you can also find parameters.
In the design of a new smart park, architects want the ratio between the lawn area and the play area to follow a specific rule. For the project to be approved, the…
In the design of a new smart park, architects want the ratio between the lawn area and the play area to follow a specific rule. For the project to be approved, the following algebraic fraction must be able to be simplified:
$$F(x) = \frac{x^2 + kx - 12}{x - 3}$$Calculate the value of the constant $k$ so that the numerator is divisible by the denominator and perform the simplification.
Fully simplify the following algebraic expression, first factorising all polynomials using algebraic identities and Ruffini's rule: $$\frac{x^4 - 13x^2 + 36}{x^3 - x^2 -…
Fully simplify the following algebraic expression, first factorising all polynomials using algebraic identities and Ruffini's rule:
$$\frac{x^4 - 13x^2 + 36}{x^3 - x^2 - 9x + 9} \times \frac{x^2 - 6x + 9}{x^2 + x - 6}$$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…
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…
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.
Identities and decomposition into partial fractions
To prove an identity, you work on one side until you reach the other (or a constant). To find $A$ and $B$ in an equality like $\dfrac{8x-11}{x^2-x-6}=\dfrac{A}{x-3}+\dfrac{B}{x+2}$, you add the right-hand side using a common denominator and set the numerators equal.
Prove, by expanding and simplifying the left-hand side of the equality, that the following expression is an identity (the result is constant): $$\frac{x + 2}{x^2 - 1}…
Prove, by expanding and simplifying the left-hand side of the equality, that the following expression is an identity (the result is constant):
$$\frac{x + 2}{x^2 - 1} \times \frac{x + 1}{x^2 - 4} - \frac{1}{x^2 - 3x + 2} = 0$$Determine the values of the real parameters $A$ and $B$ so that the following equality between fractions is valid for any value of $x$ that does not make the…
Determine the values of the real parameters $A$ and $B$ so that the following equality between fractions is valid for any value of $x$ that does not make the denominators zero:
$$\frac{8x - 11}{x^2 - x - 6} = \frac{A}{x - 3} + \frac{B}{x + 2}$$Applications: average cost and investment per customer
In economics, many quantities are quotients: average cost $=\dfrac{\text{total cost}}{\text{units}}$. With the shipping store from the beginning: $\dfrac{40x^2+200x}{x}=\dfrac{40x(x+5)}{x}=40(x+5)$ pesos per package. Back to the start: the average cost per package is $40x+200$ pesos.
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…
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: $IC(x) = \frac{C(x)}{N(x)}$.