Question

Factorise \(\frac{A}{t_{0} s}-\frac{A}{t_{0} s^{2}}\).

Short answer

Question: Factorise the expression \(\frac{A}{t_{0} s}-\frac{A}{t_{0} s^{2}}\). Answer: The factorised form of the given expression is \(\frac{A(s-1)}{t_{0} s^{2}}\).

Step-by-step solution

Find a common denominator

To find the least common denominator (LCD), we need to determine the least common multiple (LCM) of the two denominators, \(t_{0}s\) and \(t_{0}s^2\). Since \(s^2\) is a multiple of \(s\), the LCM is simply \(t_{0} s^2\).

Rewrite the expression with the common denominator

Rewrite the original expression with the common denominator of \(t_{0}s^2\). This requires multiplying the first term by \(\frac{s}{s}\): $$\frac{A}{t_{0} s}\cdot\frac{s}{s} - \frac{A}{t_{0} s^{2}}$$ This results in the following expression: $$\frac{As}{t_{0} s^{2}} - \frac{A}{t_{0} s^{2}}$$

Combine the terms

Now that both terms have the same denominator, they can be combined: $$\frac{As - A}{t_{0} s^{2}}$$

Factorise the numerator

Factorise the numerator by taking out the common factor of \(A\): $$\frac{A(s - 1)}{t_{0} s^{2}}$$ The factorised form of the given expression is \(\frac{A(s-1)}{t_{0} s^{2}}\).

Key concepts

Common Denominator

Understanding the common denominator is essential for simplifying expressions, especially when dealing with algebraic fractions. A common denominator is a shared multiple of the denominators of two or more fractions. To make the process of combining fractions possible, their denominators must be the same. Just as you can't add apples and oranges without converting one to the other, you cannot directly combine fractions with different denominators.

To find a common denominator, we often look for the least common multiple of the separate denominators, ensuring the smallest possible value to simplify calculations.

Least Common Multiple

The least common multiple (LCM) is the smallest non-zero common multiple shared between two or more numbers. In algebra, we use the LCM to find the smallest common denominator for fractions with different denominators, as seen in the above exercise.

Finding the LCM involves prime factorization or listing out multiples of the denominators to identify the smallest shared multiple. When we apply LCM to algebraic terms with variables, we also consider the variables and their powers to ensure every term in the denominator is accounted for in the LCM.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but with the inclusion of variables. Simplifying algebraic fractions involves finding a common denominator, as with numerical fractions, and then combining terms or simplifying where possible.

Working with algebraic fractions requires a careful balance of arithmetic and algebraic manipulation—adjusting the terms while respecting the rules of algebra. It's important to work methodically to prevent errors, such as prematurely canceling terms or misapplying algebraic rules.

Factorisation Techniques

Factorisation techniques enable us to write an algebraic expression as a product of its factors, making it easier to simplify and solve equations. Common methods include factoring out the greatest common factor, using special product formulas, and grouping terms methodically to find commonalities.

In our exercise, we factorise the numerator by identifying the greatest common factor and extracting it from the terms. This method is not just a mechanical process—it requires understanding of the underlying algebraic structures, and often, a bit of intuition to recognize which approach will be most efficient.