Knowunity AI

Mais

Carreiras

Programa de Criadores

Abra o App

Matérias

MathematicsMathematics4 visualizações·Atualizado May 25, 2026·6 páginas

Mastering Rational Expressions: Simplify, Solve, and Operate

Rational expressions are basically fractions with polynomials on top and... Mostrar mais

Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
1 / 6
1
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

What Are Rational Expressions?

Ever wondered what happens when you mix fractions with algebra? You get rational expressions - fractions where both the numerator and denominator are polynomials, like x2+2x−3x+5\frac{x^2+2x-3}{x+5}.

The golden rule here is that the denominator can never equal zero because dividing by zero is mathematically impossible. This creates what we call restrictions or non-permissible values - basically the values of x that would make the denominator zero.

Finding restrictions is dead simple: set the denominator equal to zero and solve. For example, with xx−4\frac{x}{x-4}, the restriction is x = 4 because that makes the bottom 4-4 = 0.

Pro tip: Always find your restrictions first - they'll be crucial when solving equations later on!

2
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Simplifying Rational Expressions

This is where factorising becomes your best mate. The process is straightforward: factorise everything, state your restrictions, then cancel common factors (not terms!).

Let's break down x2−9x2+4x+3\frac{x^2-9}{x^2+4x+3}. First, factorise the top: x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3) using difference of two squares. Then the bottom: x2+4x+3=(x+3)(x+1)x^2+4x+3 = (x+3)(x+1).

Now you can see the common factor (x+3)(x+3) and cancel it out, giving you x−3x+1\frac{x-3}{x+1} with restrictions x ≠ -3, x ≠ -1.

Warning: You can only cancel factors, never terms. Don't try cancelling the x in xx3\frac{x}{x^3} - that's mathematically wrong!

3
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Multiplying and Dividing

Good news - this bit's actually easier than adding and subtracting! For multiplication, factorise everything first, then multiply tops together and bottoms together, and cancel any common factors.

Division follows the classic "keep, change, flip" rule. Keep the first fraction as is, change the division sign to multiplication, then flip the second fraction. Just remember that when you flip a fraction, its original numerator becomes a new denominator, so you need restrictions from there too.

The key is staying organised - write down all your restrictions from every denominator (including the one you flipped) before you start cancelling.

Remember: Division is just multiplication in disguise - flip that second fraction and you're sorted!

4
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Adding and Subtracting

This is where things get properly tricky because you need a common denominator. Think of it like adding 13+14\frac{1}{3} + \frac{1}{4} - you need a common bottom first.

Here's the step-by-step: factorise all denominators, find the LCD (lowest common denominator), rewrite each fraction with the LCD, then add or subtract the numerators. Be extra careful with negative signs - use brackets like −(2x−1)=−2x+1-(2x-1) = -2x+1.

Let's try 3x+2−2x−5\frac{3}{x+2} - \frac{2}{x-5}. The LCD is (x+2)(x−5)(x+2)(x-5). Rewriting: 3(x−5)(x+2)(x−5)−2(x+2)(x+2)(x−5)\frac{3(x-5)}{(x+2)(x-5)} - \frac{2(x+2)}{(x+2)(x-5)}. This gives us 3x−15−2x−4(x+2)(x−5)=x−19(x+2)(x−5)\frac{3x-15-2x-4}{(x+2)(x-5)} = \frac{x-19}{(x+2)(x-5)}.

Top tip: When subtracting, always put brackets around the entire numerator you're subtracting to avoid sign errors!

5
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Solving Rational Equations

Now we're putting it all together! When solving equations like 5x−1−3x=12\frac{5}{x-1} - \frac{3}{x} = \frac{1}{2}, your first job is stating all restrictions (x ≠ 1, x ≠ 0).

Next, find the LCD of all terms - here it's $2xx−1x-1.MultiplyeverysingletermbythisLCDtoclearallthefractions.Aftercancelling,youget:. Multiply every single term by this LCD to clear all the fractions. After cancelling, you get: 10x - 6x−1x-1 = xx−1x-1,whichsimplifiestothequadratic, which simplifies to the quadratic x^2-5x-6=0$.

Factorising gives (x−6)(x+1)=0(x-6)(x+1)=0, so x = 6 or x = -1. Always check these solutions against your original restrictions - both are valid here since neither is 1 or 0.

Crucial step: Any solution that matches a restriction must be rejected - it's not a valid answer!

6
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Exam Success Strategy

You've got this! Here's your quick reference for exam day: simplifying means factorise, state restrictions, then cancel factors. Multiplying is factorise everything, multiply across, then cancel. Dividing is flip and multiply.

For adding/subtracting, remember the mantra: factorise denominators, find LCD, rewrite fractions, combine carefully (watch those minus signs!), then simplify. Solving equations requires restrictions first, then clear fractions with the LCD.

The most common mistakes? Cancelling terms instead of factors, forgetting restrictions, and messing up signs when subtracting. Avoid these and you're golden.

Final reminder: Restrictions aren't just busy work - they'll save you from giving impossible answers that cost marks!

Achamos que você nunca perguntaria...

O que é o assistente de IA da Knowunity?

Nosso companheiro de IA foi criado especificamente para atender às necessidades dos estudantes. Com base nos milhões de conteúdos que temos na plataforma, podemos oferecer respostas realmente relevantes e significativas. Mas não se trata apenas de respostas, o companheiro também está aqui para guiar você pelos desafios diários de aprendizado, com planos de estudo personalizados, quizzes ou conteúdos no chat e 100% de personalização com base nas suas habilidades e desenvolvimentos.

Onde posso baixar o app da Knowunity?

Pode descarregar a aplicação na Google Play Store e na Apple App Store.

Como posso receber meu pagamento? Quanto posso ganhar?

Sim, tem acesso gratuito ao conteúdo da aplicação e ao nosso companheiro de IA. Para desbloquear determinadas funcionalidades da aplicação, pode adquirir o Knowunity Pro.

Conteúdos mais populares de Mathematics

8
MathematicsMathematics

Algebra

Algebra

5th Year91
MathematicsMathematics

Algebra 2

Algebra notes focusing on the factor theorem, completing the square, -b formula, graphs of polynomials

5th Year130
MathematicsMathematics

Solving Equations

This section focuses on solving one-step and two-step linear equations to find the value of an unknown variable.

1st Year131
MathematicsMathematics

Introduction to Probability

This topic introduces basic probability concepts, including calculating the probability of simple events and understanding the difference between experimental and theoretical probability.

2nd Year81
MathematicsMathematics

Maths jc algebra

Maths jc

1st Year230
MathematicsMathematics

Natural Numbers and Integers

Students will learn about positive whole numbers, zero, and negative whole numbers, and how to add, subtract, multiply, and divide them correctly.

1st Year131
MathematicsMathematics

Differential Calculus

Calculus is a topic that comes up nearly everywhere on your maths LC. This is just starter notes that could be useful end of 5th year or start of 6th year

5th Year271
MathematicsMathematics

Arithmetic sequences and series

With examples

5th Year130

Conteúdos mais populares

9
IrishIrish

Irish oral questions and answers

Questions and answers for the leaving cert oral

5th Year4264
IrishIrish

Irish oral questions

Outline of oral questions

5th Year2055
EnglishEnglish

Key Quotes : Sive

Key Quotes and explanations: Sive

6th Year2822
IrishIrish

Iníon- le hÁine Durkin

Aine Durkin’s poem, Iníon: Themes & summary

5th Year850
IrishIrish

Irish poetry 2027

Iníon + Dínit an Bhróin

5th Year1133
IrishIrish

LC HL notes- Iníon (poem)

Includes poem in English and Irish, theme, key words & phrases

5th Year2284
IrishIrish

Gaeilge Grammar Office

All the basics you need to know on Irish grammar.

6th Year1053
EnglishEnglish

Cultural Context : Shawshank Redemption : Sive : Small Things Like These

Comparative Study : Cultural Context : Shawshank Redemption, Sive and Small Things Like These

6th Year1420
IrishIrish

An Gaeilge Aiste

Irish Language essay

6th Year1320

Não encontrou o que procurava? Explore outras matérias.

Avaliações dos nossos usuários. Eles gostaram de tudo — e você também vai gostar.

4.6/5App Store
4.7/5Google Play

O app é muito fácil de usar e bem projetado. Encontrei tudo o que estava procurando até agora e consegui aprender muito com as apresentações! Definitivamente vou usar o app para uma tarefa de classe! E, claro, também ajuda muito como inspiração.

Stefan Susuário de iOS

Este app é realmente ótimo. Tem muitos materiais de estudo e ajuda [...]. Minha matéria problemática é o francês, por exemplo, e o app tem tantas opções de ajuda. Graças a este app, eu melhorei meu francês. Eu recomendaria para qualquer pessoa.

Samantha Klichusuária de Android

Uau, estou realmente impressionado. Eu experimentei o app porque vi muitos anúncios e fiquei absolutamente maravilhado. Este app é A AJUDA que você quer para a escola e, acima de tudo, oferece muitas coisas, como treinos e resumos, que têm sido MUITO úteis para mim pessoalmente.

Annausuária de iOS

MathematicsMathematics4 visualizações·Atualizado May 25, 2026·6 páginas

Mastering Rational Expressions: Simplify, Solve, and Operate

Rational expressions are basically fractions with polynomials on top and bottom - think of them as regular fractions but with algebra thrown in. They're everywhere in maths, from solving real-world problems to advanced calculus, so getting comfortable with them now... Mostrar mais

1
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

What Are Rational Expressions?

Ever wondered what happens when you mix fractions with algebra? You get rational expressions - fractions where both the numerator and denominator are polynomials, like x2+2x−3x+5\frac{x^2+2x-3}{x+5}.

The golden rule here is that the denominator can never equal zero because dividing by zero is mathematically impossible. This creates what we call restrictions or non-permissible values - basically the values of x that would make the denominator zero.

Finding restrictions is dead simple: set the denominator equal to zero and solve. For example, with xx−4\frac{x}{x-4}, the restriction is x = 4 because that makes the bottom 4-4 = 0.

Pro tip: Always find your restrictions first - they'll be crucial when solving equations later on!

2
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

Simplifying Rational Expressions

This is where factorising becomes your best mate. The process is straightforward: factorise everything, state your restrictions, then cancel common factors (not terms!).

Let's break down x2−9x2+4x+3\frac{x^2-9}{x^2+4x+3}. First, factorise the top: x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3) using difference of two squares. Then the bottom: x2+4x+3=(x+3)(x+1)x^2+4x+3 = (x+3)(x+1).

Now you can see the common factor (x+3)(x+3) and cancel it out, giving you x−3x+1\frac{x-3}{x+1} with restrictions x ≠ -3, x ≠ -1.

Warning: You can only cancel factors, never terms. Don't try cancelling the x in xx3\frac{x}{x^3} - that's mathematically wrong!

3
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

Multiplying and Dividing

Good news - this bit's actually easier than adding and subtracting! For multiplication, factorise everything first, then multiply tops together and bottoms together, and cancel any common factors.

Division follows the classic "keep, change, flip" rule. Keep the first fraction as is, change the division sign to multiplication, then flip the second fraction. Just remember that when you flip a fraction, its original numerator becomes a new denominator, so you need restrictions from there too.

The key is staying organised - write down all your restrictions from every denominator (including the one you flipped) before you start cancelling.

Remember: Division is just multiplication in disguise - flip that second fraction and you're sorted!

4
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

Adding and Subtracting

This is where things get properly tricky because you need a common denominator. Think of it like adding 13+14\frac{1}{3} + \frac{1}{4} - you need a common bottom first.

Here's the step-by-step: factorise all denominators, find the LCD (lowest common denominator), rewrite each fraction with the LCD, then add or subtract the numerators. Be extra careful with negative signs - use brackets like −(2x−1)=−2x+1-(2x-1) = -2x+1.

Let's try 3x+2−2x−5\frac{3}{x+2} - \frac{2}{x-5}. The LCD is (x+2)(x−5)(x+2)(x-5). Rewriting: 3(x−5)(x+2)(x−5)−2(x+2)(x+2)(x−5)\frac{3(x-5)}{(x+2)(x-5)} - \frac{2(x+2)}{(x+2)(x-5)}. This gives us 3x−15−2x−4(x+2)(x−5)=x−19(x+2)(x−5)\frac{3x-15-2x-4}{(x+2)(x-5)} = \frac{x-19}{(x+2)(x-5)}.

Top tip: When subtracting, always put brackets around the entire numerator you're subtracting to avoid sign errors!

5
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

Solving Rational Equations

Now we're putting it all together! When solving equations like 5x−1−3x=12\frac{5}{x-1} - \frac{3}{x} = \frac{1}{2}, your first job is stating all restrictions (x ≠ 1, x ≠ 0).

Next, find the LCD of all terms - here it's $2xx−1x-1.MultiplyeverysingletermbythisLCDtoclearallthefractions.Aftercancelling,youget:. Multiply every single term by this LCD to clear all the fractions. After cancelling, you get: 10x - 6x−1x-1 = xx−1x-1,whichsimplifiestothequadratic, which simplifies to the quadratic x^2-5x-6=0$.

Factorising gives (x−6)(x+1)=0(x-6)(x+1)=0, so x = 6 or x = -1. Always check these solutions against your original restrictions - both are valid here since neither is 1 or 0.

Crucial step: Any solution that matches a restriction must be rejected - it's not a valid answer!

6
of 6
# Rational Expressions ## What are rational expressions? A rational expression is basically just a fraction where the numerator and the de

Cadastre-se para ver o conteúdo. É grátis!

  • Acesso a todos os documentos
  • Melhore suas notas
  • Junte-se a milhões de estudantes

Exam Success Strategy

You've got this! Here's your quick reference for exam day: simplifying means factorise, state restrictions, then cancel factors. Multiplying is factorise everything, multiply across, then cancel. Dividing is flip and multiply.

For adding/subtracting, remember the mantra: factorise denominators, find LCD, rewrite fractions, combine carefully (watch those minus signs!), then simplify. Solving equations requires restrictions first, then clear fractions with the LCD.

The most common mistakes? Cancelling terms instead of factors, forgetting restrictions, and messing up signs when subtracting. Avoid these and you're golden.

Final reminder: Restrictions aren't just busy work - they'll save you from giving impossible answers that cost marks!

Achamos que você nunca perguntaria...

O que é o assistente de IA da Knowunity?

Nosso companheiro de IA foi criado especificamente para atender às necessidades dos estudantes. Com base nos milhões de conteúdos que temos na plataforma, podemos oferecer respostas realmente relevantes e significativas. Mas não se trata apenas de respostas, o companheiro também está aqui para guiar você pelos desafios diários de aprendizado, com planos de estudo personalizados, quizzes ou conteúdos no chat e 100% de personalização com base nas suas habilidades e desenvolvimentos.

Onde posso baixar o app da Knowunity?

Pode descarregar a aplicação na Google Play Store e na Apple App Store.

Como posso receber meu pagamento? Quanto posso ganhar?

Sim, tem acesso gratuito ao conteúdo da aplicação e ao nosso companheiro de IA. Para desbloquear determinadas funcionalidades da aplicação, pode adquirir o Knowunity Pro.

Conteúdos mais populares de Mathematics

8
MathematicsMathematics

Algebra

Algebra

5th Year91
MathematicsMathematics

Algebra 2

Algebra notes focusing on the factor theorem, completing the square, -b formula, graphs of polynomials

5th Year130
MathematicsMathematics

Solving Equations

This section focuses on solving one-step and two-step linear equations to find the value of an unknown variable.

1st Year131
MathematicsMathematics

Introduction to Probability

This topic introduces basic probability concepts, including calculating the probability of simple events and understanding the difference between experimental and theoretical probability.

2nd Year81
MathematicsMathematics

Maths jc algebra

Maths jc

1st Year230
MathematicsMathematics

Natural Numbers and Integers

Students will learn about positive whole numbers, zero, and negative whole numbers, and how to add, subtract, multiply, and divide them correctly.

1st Year131
MathematicsMathematics

Differential Calculus

Calculus is a topic that comes up nearly everywhere on your maths LC. This is just starter notes that could be useful end of 5th year or start of 6th year

5th Year271
MathematicsMathematics

Arithmetic sequences and series

With examples

5th Year130

Conteúdos mais populares

9
IrishIrish

Irish oral questions and answers

Questions and answers for the leaving cert oral

5th Year4264
IrishIrish

Irish oral questions

Outline of oral questions

5th Year2055
EnglishEnglish

Key Quotes : Sive

Key Quotes and explanations: Sive

6th Year2822
IrishIrish

Iníon- le hÁine Durkin

Aine Durkin’s poem, Iníon: Themes & summary

5th Year850
IrishIrish

Irish poetry 2027

Iníon + Dínit an Bhróin

5th Year1133
IrishIrish

LC HL notes- Iníon (poem)

Includes poem in English and Irish, theme, key words & phrases

5th Year2284
IrishIrish

Gaeilge Grammar Office

All the basics you need to know on Irish grammar.

6th Year1053
EnglishEnglish

Cultural Context : Shawshank Redemption : Sive : Small Things Like These

Comparative Study : Cultural Context : Shawshank Redemption, Sive and Small Things Like These

6th Year1420
IrishIrish

An Gaeilge Aiste

Irish Language essay

6th Year1320

Não encontrou o que procurava? Explore outras matérias.

Avaliações dos nossos usuários. Eles gostaram de tudo — e você também vai gostar.

4.6/5App Store
4.7/5Google Play

O app é muito fácil de usar e bem projetado. Encontrei tudo o que estava procurando até agora e consegui aprender muito com as apresentações! Definitivamente vou usar o app para uma tarefa de classe! E, claro, também ajuda muito como inspiração.

Stefan Susuário de iOS

Este app é realmente ótimo. Tem muitos materiais de estudo e ajuda [...]. Minha matéria problemática é o francês, por exemplo, e o app tem tantas opções de ajuda. Graças a este app, eu melhorei meu francês. Eu recomendaria para qualquer pessoa.

Samantha Klichusuária de Android

Uau, estou realmente impressionado. Eu experimentei o app porque vi muitos anúncios e fiquei absolutamente maravilhado. Este app é A AJUDA que você quer para a escola e, acima de tudo, oferece muitas coisas, como treinos e resumos, que têm sido MUITO úteis para mim pessoalmente.

Annausuária de iOS