Algebra 1 - Algebraic Fractions

Introduction

• A fraction in which the numerator and denominator are algebraic expressions is called an algebraic fraction.
• $\frac{5x-2}{x^2-3x+2}$, $\frac{\sqrt{2x-5}}{3x}$, etc are some examples of algebraic fractions.

Algebraic Fraction vs Rational Fractions

• The difference between algebraic fractions and rational fractions is described below.
• Algebraic Fractions:
  • In algebraic fractions, both the numerator and denominator are algebraic expressions.
  • Examples: $\frac{x}{\sqrt{x}+2}$, $3x^{\frac{2}{3}}+7$, $\frac{x^2-16}{x+2}$, etc
• Rational Fractions:
  • In rational fractions, both the numerator and denominator are polynomials.
  • Examples: $3x^2-21$, $\frac{x-1}{x^2-4x+3}$, etc

Adding and Subtracting Algebraic Fractions

• Two algebraic fractions can only be added or subtracted if they have a common (same) denominator.
• If the two algebraic fractions to be added don't have a common denominator, we need to first modify the algebraic fractions in such a way that equivalent algebraic fractions have common denominators.

For simplicity, we classify the addition and subtraction of algebraic expressions into two cases. 

1. When algebraic fractions have a common (same) denominator:

• Keep the denominator as it is, and then add or subtract the numerators.
•  Simplify (cancel out common factors) the resulting expression if possible.

2. When algebraic fractions have different denominators:

• Find the LCM (least common multiple) of all the denominators.
• Modify the given algebraic fractions into an equivalent algebraic fraction with the same denominator.
• Simplify the resulting algebraic fractions if possible.

Multiplying Algebraic Fractions

To multiply two algebraic fractions, follow the steps below:

• Factor the numerators and denominators.
• Simplify the expression by canceling out the common factors.
• Multiply the numerator of one fraction by the numerator of the other.
• Multiply the denominator of one fraction by the denominator of the other.

Dividing Algebraic Fractions

To divide the two algebraic fractions, follow the steps below:

• Multiply the first algebraic fraction with the reciprocal of the second algebraic fraction.
• Simplify the expression by canceling out the common factors.

Solved Examples

Example 1: Simplify $\frac{1-x}{x^2+2x+1}+\frac{2x}{x^2+2x+1}$.

Solution: $\frac{1-x}{x^2+2x+1}+\frac{2x}{x^2+2x+1}$$=\frac{1-x+2x}{x^2+2x+1}$$=\frac{x+1}{x^2+2x+1}$$=\frac{\left(x+1\right)}{{\left(x+1\right)}^2}$$=\frac{1}{x+1}$

 

Example 2: Simplify $\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{1}{\sqrt{x}-1}$.

Solution: $\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{1}{\sqrt{x}-1}$$=\frac{\sqrt{x}·\left(\sqrt{x}-1\right)+\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)·\left(\sqrt{x}-1\right)}$$=\frac{x-\sqrt{x}+\sqrt{x}+1}{{\left(\sqrt{x}\right)}^2-{\left(1\right)}^2}$$=\frac{x+1}{x-1}$

 

Example 3: Simplify $\frac{2x}{x-2}-\frac{x-1}{x+2}$.

Solution: $\frac{2x}{x-2}-\frac{x-1}{x+2}$$=\frac{2x·\left(x+2\right)-\left(x-2\right)·\left(x-1\right)}{\left(x-2\right)·\left(x+2\right)}$$=\frac{2x^2+4x-\left(x^2-3x+2\right)}{\left(x-2\right)·\left(x+2\right)}$$=\frac{2x^2+4x-x^2+3x-2}{\left(x-2\right)·\left(x+2\right)}$$=\frac{x^2+7x-2}{x^2-4}$

 

Example 4: Simplify $\frac{x^5}{18y^3}\times \frac{9y^7}{2x}$.

Solution: $\frac{x^5}{18y^3}\times \frac{9y^7}{2x}$$=\frac{x^4}{2}\times \frac{y^4}{2}$$=\frac{x^4{y}^4}{4}$

 

Example 5: Simplify $\frac{2x^2-4x}{x^2-3x+2}\times \frac{x-1}{2x+2}$.

Solution: $\frac{2x^2-4x}{x^2-3x+2}\times \frac{x-1}{2x+2}$$=\frac{2x\left(x-2\right)}{\left(x-2\right)\left(x-1\right)}\times \frac{\left(x-1\right)}{2\left(x+1\right)}$$=\frac{x}{1}\times \frac{1}{\left(x+1\right)}$$=\frac{x}{x+1}$

 

Example 6: Simplify $\frac{25x^3}{13y^{{\displaystyle \frac{3}{2}}}}÷\frac{100x^5}{26y^{\frac{5}{2}}}$.

Solution: $\frac{25x^3}{13y^{{\displaystyle \frac{3}{2}}}}÷\frac{100x^5}{26y^{\frac{5}{2}}}$$=\frac{25x^3}{13y^{{\displaystyle \frac{3}{2}}}}\times \frac{26y^{{\displaystyle \frac{5}{2}}}}{100x^5}$$=\frac{2y^{{\displaystyle \frac{5}{2}-\frac{3}{2}}}}{4x^2}$$=\frac{y}{2x^2}$

 

Example 7: Simplify $\frac{2x^2-8}{x-2}÷\frac{x^2+4x+4}{x+2}$.

Solution: $\frac{2x^2-8}{x-2}÷\frac{x^2+4x+4}{x+2}$$=\frac{2x^2-8}{x-2}\times \frac{x+2}{x^2+4x+4}$$=\frac{2\left(x^2-4\right)}{\left(x-2\right)}\times \frac{\left(x+2\right)}{{\left(x+2\right)}^2}$$=\frac{2\left(x+2\right)\left(x-2\right)}{\left(x-2\right)}\times \frac{\left(x+2\right)}{{\left(x+2\right)}^2}$$=2$

Cheat Sheet

• Algebraic fractions can be added or subtracted only if they have the same denominators.
• Suppose the algebraic fractions to be added or subtracted do not have a common denominator. In that case, both must be first converted into equivalent algebraic fractions with common denominators, and then an addition or subtraction operation be performed.
• In the multiplication of algebraic fractions, we multiply the numerator of one bit with the numerator of the other and similarly multiply the denominator of one fraction with that of another. Then, finally, we simplify to get the result.
• In the division of algebraic fractions, we multiply the first fraction with the reciprocal of the second fraction, followed by their simplification.

Blunder Areas

• All rational fractions are algebraic fractions, but not all algebraic fractions are rational fractions.