Algebraic Fractions Arithmetic

An algebraic fraction is a fraction in which the numerator and denominator are both polynomials. For example, the fraction (2x+3)/(x-4) is an algebraic fraction.

To add or subtract algebraic fractions, we must first ensure that the denominators are the same. If the denominators are not already the same, we can find a common denominator by multiplying the two denominators together. Then, we can rewrite each fraction with the common denominator and proceed with the addition or subtraction as we would with any other fractions. For example, to add the fractions (2x+3)/(x-4) and (x+1)/(x-4), we can find a common denominator of (x-4)^2 and rewrite the fractions as (2x+3)/(x-4) + (x+1)/(x-4) = (2x+3)(x-4) + (x+1)(x-4)/(x-4)^2. We can then simplify the expression to get our final answer.

To multiply algebraic fractions, we simply multiply the numerators and denominators as we would with any other fractions. For example, to multiply the fractions (2x+3)/(x-4) and (x+1)/(x-4), we would get the product (2x+3)(x+1)/(x-4)^2.

To divide algebraic fractions, we invert the second fraction and then multiply the two fractions as we would with any other fractions. For example, to divide the fraction (2x+3)/(x-4) by the fraction (x+1)/(x-4), we would invert the second fraction to get (2x+3)/(x-4) / (x+1)/(x-4) = (2x+3)/(x-4) * (x-4)/(x+1). We can then simplify the expression to get our final answer.

Algebraic fractions can be simplified by factoring the numerator and denominator and canceling common factors. For example, the fraction (2x+3)/(x-4) can be simplified by factoring the numerator and denominator to get (2x+3)/(x-4) = (2x+3)/(x-4) * (x+4)/(x+4) = (2x+3)(x+4)/(x-4)(x+4) = (2x^2+10x+12)/(x^2-16).