Find your next lesson

Published Calculus and Precalculus lessons.

Start with a lesson, or search by title, code, or skill.

Esc or Close to return

Course menu
Course home

Grade 11 Precalculus · Lesson 1.4

Rational Expressions

Factor first. Protect the restrictions. Then choose the operation. Connect simplifying, operations, LCDs, compound fractions, rational exponents, and rationalizing into one reliable system.
10 learning scenesInteractiveSame lesson material

Choose a section to read or explore.

Explanation

1. Rational Expressions & Domain Restrictions

A rational expression is a quotient of two polynomials. The denominator cannot equal zero. Record restrictions from the original denominator before cancelling.

Interactive worked example · restrictions survive cancellation
x2+5x+6x2+8x+15\dfrac{x^2+5x+6}{x^2+8x+15}

Record restrictions from the original denominator before cancelling anything.

Your turn · Factor the original denominator

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Routine: Factor completely → state restrictions → cancel common factors → simplify.

Interactive lab

Restriction & Cancellation Visualizer

The cancelled factor disappears from the formula, but its restriction stays in the domain.

Your expression

x2−4x2+x−6\dfrac{x^2-4}{x^2+x-6}

Checkpoint 1 of 2. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Which values are excluded before cancelling?

x2+x−6=(x+3)(x−2)x^2+x-6=(x+3)(x-2)x≠−3,2x\ne-3,2

2. Which final form preserves the original domain?

(x−2)(x+2)(x+3)(x−2)\dfrac{(x-2)(x+2)}{(x+3)(x-2)}x+2x+3,x≠−3,2\dfrac{x+2}{x+3},\quad x\ne-3,2
Which values are excluded before cancelling?

Your turn

What must you do before cancelling a common factor?

Your turn

Simplify x2−9x2+x−6\frac{x^2-9}{x^2+x-6} while preserving the original domain.
Explanation

2. Multiplying & Dividing

Multiply: factor first, cancel common factors, then multiply. Divide: Keep – Change – Flip, then factor and cancel. Never cancel terms joined by addition or subtraction.

Example · multiply after factoring
x2−9x2+5x+6⋅x+2x−3\dfrac{x^2-9}{x^2+5x+6}\cdot\dfrac{x+2}{x-3}

Factor every polynomial before cancelling.

Your turn · Factor

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Example · divide using Keep–Change–Flip
x2−4x2+x−6÷x+2x−2\dfrac{x^2-4}{x^2+x-6}\div\dfrac{x+2}{x-2}

Change division to multiplication by the reciprocal before cancelling.

Your turn · Keep–Change–Flip

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

What comes immediately after changing division to multiplication?

Keep the original domain. Multiplication example: x≠−3,−2,3x\ne-3,-2,3. Division example: x≠−3,2,−2x\ne-3,2,-2; the divisor itself must be nonzero.

Explanation

3. Adding & Subtracting · Think LCD

Factor denominators first. Build the LCD, rewrite each fraction, combine numerators, then simplify.

Example · unlike linear denominators
3x−1+xx+2\dfrac{3}{x-1}+\dfrac{x}{x+2}

Build the LCD first, then combine the numerators.

Your turn · LCD

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Example · factor before choosing the LCD
2x2−4−1x2+x−6\dfrac{2}{x^2-4}-\dfrac{1}{x^2+x-6}

Factor both denominators first so the LCD is visible.

Your turn · Factor denominators

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive lab

LCD Factor Builder

Factor each denominator. Include each required factor at its greatest needed power.

Your expression

x−2x-2 and x+3x+3

Checkpoint 1 of 1. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Choose the LEAST common denominator.

x−2x-2x+3x+3(x−2)(x+3)(x-2)(x+3)
Choose the LEAST common denominator.

Your turn

What determines the LCD?
Explanation

4. Compound Fractions · Preferred LCD Method

Key takeaway

Mr. Azzam's preferred method

Find one LCD for all small denominators and multiply the entire numerator and entire denominator by it. This has priority over simplify-then-divide.

xy+11−yx\frac{\frac{x}{y}+1}{1-\frac{y}{x}}

Here the LCD is xy. Multiply both large parts by it:

xy(xy+1)xy(1−yx)=x(x+y)y(x−y)\frac{xy\left(\frac{x}{y}+1\right)}{xy\left(1-\frac{y}{x}\right)}=\frac{x(x+y)}{y(x-y)}
Example · three different small denominators
1x+1y1x−1z\dfrac{\frac1x+\frac1y}{\frac1x-\frac1z}

Find one LCD for every small denominator in the entire compound fraction.

Your turn · LCD

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

What is the preferred first move for a compound fraction?
Interactive lab

LCD Builder

Clear the small fractions in one organized move. Preserve every original restriction.

Your expression

xy+11−yx\dfrac{\frac{x}{y}+1}{1-\frac{y}{x}}

Checkpoint 1 of 3. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Choose the LCD of the small fractions.

xyxyxy(xy+1)=x2+xyxy(\frac{x}{y}+1)=x^2+xyxy(1−yx)=xy−y2xy(1-\frac{y}{x})=xy-y^2

2. Choose the simplified expression.

xy(xy+1)=x2+xyxy(\frac{x}{y}+1)=x^2+xyxy(1−yx)=xy−y2xy(1-\frac{y}{x})=xy-y^2x(x+y)y(x−y)\dfrac{x(x+y)}{y(x-y)}

3. Which original domain must be retained?

x≠0, y≠0, x≠yx\ne0,\ y\ne0,\ x\ne y
Choose the LCD of the small fractions.

Keep the original domain. For the first compound fraction, x≠0x\ne0, y≠0y\ne0 and x≠yx\ne y. For the three-denominator example, x,y,z≠0x,y,z\ne0 and z≠xz\ne x. Check the large denominator as well as every small one.

Explanation

5. Rational & Negative Exponents Inside Fractional Expressions · High Priority

The goal here is not to practise exponent laws in isolation. The exponents appear inside a fractional expression, and the task is to simplify the whole expression.

Example A · rational exponents in a quotient
x3/2+2x1/2x1/2\dfrac{x^{3/2}+2x^{1/2}}{x^{1/2}}

Look for a common power in the numerator before simplifying the fraction.

Your turn · Factor the numerator

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Example B · negative exponents inside a fraction
x−1/2+3x−3/2x−3/2\dfrac{x^{-1/2}+3x^{-3/2}}{x^{-3/2}}

Do not simplify each negative exponent separately. Factor the lowest power from the numerator.

Your turn · Lowest power

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Example C · rational exponents in numerator and denominator
x5/2−4x1/2x3/2\dfrac{x^{5/2}-4x^{1/2}}{x^{3/2}}

Factor first; then simplify the common power across the fraction.

Your turn · Factor the numerator

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Example D · clear a negative fractional power
x−1/2+x1/2x−1/2−x1/2\dfrac{x^{-1/2}+x^{1/2}}{x^{-1/2}-x^{1/2}}

Multiply the entire numerator and denominator by the power that clears the negative exponent.

Your turn · Choose the clearing power

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive lab

Power-Factor Visualizer

Factor the lowest common power, simplify, and retain the original real domain.

Your expression

x−1/2+3x−3/2x−3/2\dfrac{x^{-1/2}+3x^{-3/2}}{x^{-3/2}}

Checkpoint 1 of 3. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Choose the common lowest power.

x−3/2x^{-3/2}x−3/2(x+3)x−3/2\dfrac{x^{-3/2}(x+3)}{x^{-3/2}}

2. Choose the simplified expression.

x−3/2(x+3)x−3/2\dfrac{x^{-3/2}(x+3)}{x^{-3/2}}x+3x+3

3. Which original real domain remains?

x>0x>0
Choose the common lowest power.

Quick Check

Check the idea before moving on.

What is the best first step?
x7/3+2x4/3x4/3\frac{x^{7/3}+2x^{4/3}}{x^{4/3}}

Your turn

After simplifying x3/2+2x1/2x1/2\frac{x^{3/2}+2x^{1/2}}{x^{1/2}} to x+2x+2, which real inputs remain allowed?

Keep the original domain. Examples A, B and C require x>0x>0. Example D also excludes x=1x=1. Simplified formulas retain these original real-domain restrictions.

Explanation

6. Rationalizing the Denominator

For a binomial denominator containing a radical, multiply by its conjugate. The conjugate works through the difference-of-squares identity.

11+2⋅1−21−2=2−1\frac{1}{1+\sqrt2}\cdot\frac{1-\sqrt2}{1-\sqrt2}=\sqrt2-1
Example · binomial radical denominator
43−5\dfrac{4}{3-\sqrt5}

Use the conjugate of the entire denominator.

Your turn · Choose the conjugate

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

Which factor rationalizes the denominator 5−35-\sqrt{3}?
Explanation

7. Rationalizing the Numerator

The same conjugate idea applies when the radical expression is in the numerator.

5−32⋅5+35+3=−25+3\frac{\sqrt5-3}{2}\cdot\frac{\sqrt5+3}{\sqrt5+3}=-\frac{2}{\sqrt5+3}
Example · radical binomial in the numerator
x−3x−9\dfrac{\sqrt{x}-3}{x-9}

Use the numerator conjugate, then look for a factor that cancels.

Your turn · Multiply by the numerator conjugate

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

Which conjugate is used for the numerator x−4\sqrt{x}-4?
Interactive lab

Conjugate Lab

Choose a target, its conjugate and the equivalent result. Multiply by the conjugate divided by itself.

Your expression

11+2\dfrac1{1+\sqrt2}

Checkpoint 1 of 3. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Which part is the requested rationalization target?

Target: denominator.

2. Choose its conjugate.

1−21-\sqrt2(1+2)(1−2)=1−2=−1(1+\sqrt2)(1-\sqrt2)=1-2=-1

3. Choose the equivalent result.

(1+2)(1−2)=1−2=−1(1+\sqrt2)(1-\sqrt2)=1-2=-12−1\sqrt2-1
Which part is the requested rationalization target?

Keep the original domain. The variable numerator-conjugate example requires x≥0x\ge0 and x≠9x\ne9 even after cancellation.

Explanation

8. Alternative Method · Simplify Then Divide

This method is valid, but it is the secondary method in this lesson. First simplify the large numerator into one fraction, simplify the large denominator into one fraction, then multiply by the reciprocal.

Use it to understand the structure. For efficient work with several small fractions, use the LCD method first.

Quick Check

Check the idea before moving on.

A compound fraction contains four small denominators. Which approach has priority?
Practice

9. Mastery Checkpoint

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Restrictions & simplifying

Multiplying & dividing rational expressions

LCD for addition & subtraction

Compound fractions · LCD method

Rational & negative rational exponents

Rationalizing denominator

Rationalizing numerator

Rate all skills to prepare your personalized practice path.
Common mistake

10. Exam Strategy & Common Mistakes

  • State restrictions before cancelling.
  • Cancel factors, not terms.
  • Factor denominators before finding the LCD.
  • For compound fractions, prefer clearing all small fractions with one LCD.
  • Track rational and negative exponents carefully.
  • Use a conjugate for a two-term radical denominator or numerator.

Keep your unfinished answers?

You have answers you have not submitted. A saved draft stays in this browser only; it is not a submitted result.