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MathematicsMathematics4 閲覧数·更新日 May 25, 2026·6 ページ

Mastering Rational Expressions: Simplify, Solve, and Operate

Rational expressions are basically fractions with polynomials on top and... もっと見る

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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!

そんなこと聞いてくれるのを待ってたよ...

KnowunityのAIコンパニオンとは？

KnowunityのAIコンパニオンは学生向けに設計されたAIツールで、単なる答えを提供するだけではありません。数百万のKnowunityリソースを基に構築され、関連する情報、個別の学習プラン、クイズ、コンテンツをチャット内で直接提供し、あなたの個別の学習過程に適応します。

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Knowunityは本当に無料ですか？

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このアプリはとても使いやすくて、デザインも良いです。今のところ探していたものは全て見つかったし、プレゼン資料からもたくさん学べました！絶対に課題でも使いたいと思います！もちろん、アイデアを得るのにもすごく役立ちます。

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AnnaiOSユーザー

MathematicsMathematics4 閲覧数·更新日 May 25, 2026·6 ページ

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... もっと見る

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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

サインアップしてコンテンツを見よう。無料だよ！

  • 全ドキュメントへのアクセス
  • 成績アップ
  • 数百万人の学生と一緒に学習

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!

そんなこと聞いてくれるのを待ってたよ...

KnowunityのAIコンパニオンとは？

KnowunityのAIコンパニオンは学生向けに設計されたAIツールで、単なる答えを提供するだけではありません。数百万のKnowunityリソースを基に構築され、関連する情報、個別の学習プラン、クイズ、コンテンツをチャット内で直接提供し、あなたの個別の学習過程に適応します。

Knowunityアプリはどこでダウンロードできますか？

Google Play StoreとApple App Storeからアプリをダウンロードできます。

Knowunityは本当に無料ですか？

その通り！学習コンテンツへの無料アクセス、仲間の学生とのつながり、そして即座のサポートを手のひらで楽しもう。

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生徒たちが愛用中 — あなたもきっと気に入るはず。

4.6/5App Store
4.7/5Google Play

このアプリはとても使いやすくて、デザインも良いです。今のところ探していたものは全て見つかったし、プレゼン資料からもたくさん学べました！絶対に課題でも使いたいと思います！もちろん、アイデアを得るのにもすごく役立ちます。

Stefan SiOSユーザー

このアプリは本当に素晴らしいです。学習ノートやサポート資料がとても豊富で[...]。例えば、私の苦手科目はフランス語なんですが、このアプリにはサポートオプションがたくさんあります。このアプリのおかげでフランス語が上達しました。誰にでもおすすめしたいです。

Samantha KlichAndroidユーザー

すごい、本当に驚いた。広告で何度も見かけたからアプリを試してみたら、めちゃくちゃ感動した。このアプリは学校で欲しかった「まさにこれ！」って感じのサポートで、特に練習問題や要点まとめみたいな機能がたくさんあって、個人的にすごく助かってる。

AnnaiOSユーザー