Simplifying Algebraic and Rational Expressions for the ESAT

Simplifying expressions involving sums, products and powers

Algebraic expressions can be simplified using a systematic set of operations. The standard approach involves a specific sequence of steps:

1. Simplify any terms containing fractions by cancelling common factors.
2. Collect like terms whenever possible.
3. Identify and extract common factors from two or more terms.
4. If no common factors are immediately apparent, multiply out any brackets and then repeat the initial steps of simplifying fractions and collecting like terms.

Worked Example: Basic Simplification

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 ($3x^2y$ with $9x^2y$, and $4x$ with $18x$ and $2x$): $12x^2y + 24x$

Finally, take out the common factor $12x$: $12x(xy + 2)$

Worked Example: Factorising in Pairs

Sometimes an expression has no common factor across all terms, but can be simplified by grouping.

Simplify: $3x + 20y + 12 + 5xy$

First, rearrange the terms to group related variables: $20y + 5xy + 3x + 12$

Now, factorise the terms in pairs: $5y(4 + x) + 3(x + 4)$

Notice that $(x + 4)$ and $(4 + x)$ are identical. Since $(x + 4)$ is now a common factor of both parts, we can complete the factorisation: $(x + 4)(5y + 3)$

Cancelling in Rational Expressions

Rational expressions, which are fractions where the numerator and denominator are algebraic expressions, are simplified similarly to numerical fractions. You divide both the numerator and the denominator by the same number, term, or expression until no further division is possible.

Worked Example: Stepwise Cancelling

Simplify $\frac{6x^3 y^4}{15x^5 y}$

First, divide the coefficients by their highest common factor, 3: $\frac{2x^3 y^4}{5x^5 y}$

Next, divide the numerator and denominator by $x^3$: $\frac{2y^4}{5x^2 y}$

Finally, divide both by $y$: $\frac{2y^3}{5x^2}$

Rational expressions involving sums or differences

If a rational expression contains additions or subtractions in the numerator or denominator, you must factorise these sums or differences before attempting to cancel. You cannot cancel individual terms across a plus or minus sign.

Worked Example: Factorising before Cancelling

Simplify $\frac{4x^2 y + 8xy^2}{3x + 6y}$

First, identify the common factors in the numerator ($4xy$) and the denominator ($3$): $\frac{4xy(x + 2y)}{3(x + 2y)}$

Now, cancel the common expression $(x + 2y)$ from both top and bottom: $\frac{4xy}{3}$

Rational expressions involving quadratics

When dealing with quadratic expressions within a fraction, factorise the quadratics fully before trying to simplify the expression.

Worked Example: Quadratic Simplification

Simplify $\frac{6x^2 + 19x + 10}{4x^2 - 25}$

First, factorise the numerator and the denominator. Note that the denominator is a difference of two squares: $\frac{(3x + 2)(2x + 5)}{(2x + 5)(2x - 5)}$

Now, cancel the common factor $(2x + 5)$: $\frac{3x + 2}{2x - 5}$

Adding and subtracting rational expressions

To add or subtract algebraic rational expressions, they must be placed over a common denominator, following the same rules as numerical fractions. The most efficient common denominator is the lowest common multiple (LCM) of the existing denominators, although the product of the denominators can also be used.

In general: $\frac{x}{a} + \frac{y}{b} = \frac{xb + ya}{ab}$

Using the LCM reduces the amount of cancelling required in the final answer. For instance, the LCM of $6x$ and $15y$ is $30xy$, while their product is $90xy$. Similarly, the LCM of $6x^4$ and $15x^2y$ is $30x^4y$.

Worked Example: Subtraction with Denominator Manipulation

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 find a common denominator, notice that $(1 - p)$ is $-(p - 1)$. Multiply the numerator and denominator of the second fraction by $-1$: $\frac{p + 1}{p(1 - p)} + \frac{2}{1 - p}$

Now, the common denominator is $p(1 - p)$. Multiply the second term by $\frac{p}{p}$: $\frac{(p + 1) + 2p}{p(1 - p)}$

Simplify the numerator: $\frac{3p + 1}{p(1 - p)}$

Multiplying and dividing rational expressions

Algebraic rational expressions follow the standard rules for fraction multiplication and division:

Multiplication: $\frac{x}{a} \times \frac{y}{b} = \frac{xy}{ab}$

Division: $\frac{x}{a} \div \frac{y}{b} = \frac{x}{a} \times \frac{b}{y} = \frac{xb}{ay}$