Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Like regular fractions, algebraic fractions consist of a numerator (the top part) and a denominator (the bottom part), separated by a horizontal line or a division symbol.

They are used to represent ratios, proportions, or divisions involving variables, constants, and coefficients. For example, \frac{(2x + 3)}{(x - 1)} is an algebraic fraction.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, factor the numerator and denominator and cancel out any common factors. This process reduces the fraction to its simplest form.

For example, consider the algebraic fraction \frac{(6x^2 + 3x)}{(3x)}:

Step 1: Factor the numerator and denominator:

\frac{(3x(2x + 1))}{(3x)}

Step 2: Cancel out the common factor 3x:

(2x + 1)

Operations with Algebraic Fractions

Addition and subtraction

To add or subtract algebraic fractions, find a common denominator, then add or subtract the numerators.

For example, to add \frac{(2x + 3)}{(x - 1)} and \frac{(x + 2)}{(x + 1)}:

  • Step 1: Find a common denominator, which is (x - 1)(x + 1)
  • Step 2: Add the numerators: (2x + 3)(x + 1) + \frac{(x + 2)}{(x - 1)}
  • Step 3: Simplify the result: \frac{(2x^2 + 5x + 3)}{(x^2 - 1)}

Multiplication

To multiply algebraic fractions, multiply the numerators together and the denominators together. Then simplify the result, if possible.

For example, to multiply \frac{(2x + 3)}{(x - 1)} by \frac{(x + 1)}{(3x)}:

Step 1: Multiply the numerators:

(2x + 3)(x + 1)

Step 2: Multiply the denominators:

4(x - 1)(3x)

Step 3: Simplify the result:

\frac{(2x^2 + 5x + 3)}{(3x^2 - 3x)}

Division

To divide algebraic fractions, invert the divisor (flip the numerator and denominator), then multiply the fractions and simplify the result.

For example, to divide \frac{(2x + 3)}{(x - 1)} by \frac{(x + 2)}{(x + 1)}:

Step 1: Invert the divisor:

\frac{(x + 1)}{(x + 2)}

Step 2: Multiply the fractions:

\frac{((2x + 3)}{(x - 1))} \times \frac{(x + 1)}{(x + 2)}

Step 3: Simplify the result:

\frac{(2x^2 + x + 3)}{(x^2 + x - 2)}

Solving Equations with Algebraic Fractions

When solving equations with algebraic fractions, clear fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. Then, simplify and solve the resulting equation.

For example, solve the equation \frac{(2x)}{(x - 1)} = \frac{(x + 3)}{(x + 1)}:

Step 1: Multiply both sides by the LCM, (x - 1)(x + 1):

(2x)(x + 1) = (x + 3)(x - 1)

Step 2: Simplify and solve:

2x^2 + 2x = x^2 - x - 3

x^2 + 3x + 3 = 0

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