Simplifying and Operating on Rational Expressions

Simplifying Expressions Involving Sums, Products and Powers

Algebraic expressions can be simplified using several systematic steps. The standard approach involves simplifying any fractional terms through cancellation, collecting like terms where they exist, and identifying common factors across terms. If no immediate common factors are visible, you should multiply out any brackets and then repeat the process of collecting like terms and simplifying fractions. For example, consider the expression $3x^2y + 4x + 9x(xy + 2) + 2x$. First, multiply out the bracket: $3x^2y + 4x + 9x^2y + 18x + 2x$. Next, collect the like terms to get $12x^2y + 24x$. Finally, you can take out a common factor of $12x$ to reach the fully simplified form: $12x(xy + 2)$.

In some cases, you may need to rearrange terms to facilitate factorisation in pairs. For instance, in the expression $3x + 20y + 12 + 5xy$, there are no common factors for all four terms. By rearranging to $20y + 5xy + 3x + 12$, we can factorise the first two terms and the last two terms separately: $5y(4 + x) + 3(x + 4)$. Since $(x + 4)$ is now a common factor, the expression simplifies to $(x + 4)(5y + 3)$.

Cancelling in Rational Expressions

Rational expressions are simplified in the same manner as numerical fractions: by dividing both the numerator and denominator by the same number, term, or algebraic expression until no further reduction is possible. Consider the simplification of $(6x^3 y^4)/(15x^5 y)$. First, divide the numerical coefficients 6 and 15 by their highest common factor, 3, giving $\frac{2x^3 y^4}{5x^5 y}$. Next, divide the numerator and denominator by $x^3$, resulting in $\frac{2y^4}{5x^2 y}$. Finally, divide by $y$ to obtain the simplest form: $\frac{2y^3}{5x^2}$.

Rational Expressions with Sums or Differences

If a rational expression contains sums or differences, you must factorise these components before attempting to cancel. For example, to simplify $\frac{4x^2 y + 8xy^2}{3x + 6y}$, we first identify common factors in the numerator and denominator. The numerator factorises to $4xy(x+2y)$ and the denominator factorises to $3(x+2y)$. The expression becomes $\frac{4xy(x+2y)}{3(x+2y)}$. We can then cancel the common binomial term $(x+2y)$ to leave $\frac{4xy}{3}$.

Rational Expressions Involving Quadratics

When dealing with quadratic expressions within a fraction, factorise them completely to find common factors. Take the expression $\frac{6x^2 + 19x + 10}{4x^2 - 25}$. The numerator is a quadratic that factorises to $(3x+2)(2x+5)$. The denominator is a difference of two squares, which factorises to $(2x+5)(2x-5)$. By rewriting the fraction as $\frac{(3x+2)(2x+5)}{(2x+5)(2x-5)}$, we can cancel the $(2x+5)$ terms, leaving the simplified result $\frac{3x+2}{2x-5}$.

Adding and Subtracting Rational Expressions

To add or subtract algebraic fractions, you must place them over a common denominator, just as with numerical fractions. While any common multiple of the denominators will work, using the lowest common multiple (LCM) is advantageous because it often results in an expression that requires less subsequent simplification. The general rule is $\frac{x}{a} + \frac{y}{b} = \frac{xb + ya}{ab}$. For example, to simplify $\frac{p+1}{p-p^2} - \frac{2}{p-1}$, first factorise the denominators: $\frac{p+1}{p(1-p)} - \frac{2}{p-1}$. To create a common denominator, we can multiply the second fraction by $-1$ in both the numerator and denominator to change $(p-1)$ into $(1-p)$. This gives $\frac{p+1}{p(1-p)} + \frac{2}{1-p}$. Now, using $p(1-p)$ as the common denominator, we get $\frac{(p+1) + 2p}{p(1-p)}$, which simplifies to $\frac{3p+1}{p(1-p)}$.

Multiplying and Dividing Rational Expressions

Multiplication and division of algebraic fractions follow these general rules: $\frac{x}{a} \times \frac{y}{b} = \frac{xy}{ab}$ and $\frac{x}{a} \div \frac{y}{b} = \frac{x}{a} \times \frac{b}{y} = \frac{xb}{ay}$. In division, you must invert the divisor (the second fraction) and then multiply.