Algebraic fractions
An algebraic fraction such as $\dfrac{x^2-9}{x^2+6x+9}$ is handled in exactly the same way as $\dfrac{10}{15}$: to simplify it, add it or divide it you need to factorise first, just as with numbers you need to know their factors. Everything covered in Polynomials is applied here in a single block of working.
Simplifying by factorising the numerator and denominator
Simplifying an algebraic fraction means cancelling the factors that the numerator and denominator have in common. The essential step, always, is to factorise both of them first.
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:
$$\frac{4x^2 + 8x}{2x^2 - 6x}$$Multiplying and dividing algebraic fractions
Just as with numerical fractions: to multiply, multiply numerator by numerator and denominator by denominator; to divide, multiply by the reciprocal fraction. Factorising before multiplying almost always lets you simplify as you go.
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?
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.
Addition and subtraction: the lowest common multiple of the denominators
With the same denominator, you add the numerators without touching the denominator. With different denominators, you need a common denominator, which is found by factorising each denominator and taking the lowest common multiple.
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}$$Calculate the following product of algebraic fractions and simplify the result before multiplying out the terms if possible: $$\frac{x + 3}{5} \cdot \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} \cdot \frac{10}{x^2 - 9}$$In a study on the spread of news on social media, it has been determined that users' response time (in seconds) depends on two variables combined in the following…
In a study on the spread of news on social media, it has been determined that users' response time (in seconds) depends on two variables combined in the following operation:
$$\frac{3x}{x-2} - \frac{x-1}{x+2} - \frac{8}{x^2-4}$$Perform the operation by putting the fractions over a common denominator and simplify the result to obtain a single algebraic fraction that models this behaviour.
Combined operations and complex fractions
When sums, differences, products and brackets are mixed together, you work out what is inside the brackets first and then carry on step by step, just as with numbers. A complex fraction — one with fractions inside its numerator or denominator — is dealt with by first reducing each level to a single fraction, and dividing at the end.
An urban logistics company uses drones to deliver packages. The base area of the transport boxes is rectangular and its dimensions, in decimetres, are given by the…
An urban logistics company uses drones to deliver packages. The base area of the transport boxes is rectangular and its dimensions, in decimetres, are given by the expressions shown in the image. Calculate the simplified expression representing the total area of the box's base by multiplying both dimensions.
The dimensions are $Largo = \frac{x^2 - 1}{x}$ and $Ancho = \frac{x+2}{x-1}$. Calculate the area of the rectangle by simplifying the product of both fractions.
Fully simplify the following algebraic expression, first factorising all polynomials using algebraic identities and Ruffini's rule: $$\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:
$$\frac{x^4 - 13x^2 + 36}{x^3 - x^2 - 9x + 9} \cdot \frac{x^2 - 6x + 9}{x^2 + x - 6}$$