Calculus & Algebra

Calculus requires strong algebra skills. Finding derivatives, integrals, and their applications often involves simplifying expressions, factoring, and manipulating equations accurately. Practicing algebra regularly ensures a solid foundation, making it easier to focus on the concepts of calculus rather than struggling with basic algebraic operations.

Factorising

The are several methods of factorisation, including common factors, grouping, difference of squares, and quadratic trinomials. The common factor method involves taking out the greatest common factor (GCF) from terms in an expression. The grouping is used when terms can be rearranged and factored in pairs. The difference of squares method applies when an expression follows the form a² - b² = (a - b)(a + b). Factoring quadratic trinomials like ax² + bx + c requires rewriting the expressions as the product of two binomial factors.

In many cases, more than one method may be needed to fully factorise an expression. For example, a quadratic may first require taking out a common factor before using the quadratic trinomial method. Developing flexibility in applying different techniques is essential for mastering algebra and preparing for higher-level topics like calculus. Flexibility comes with practice.

Algebraic Fractions

Numerical fractions and algebraic fractions are very similar because they both represent parts of a whole. A numerical fraction has numbers in the numerator (top) and denominator (bottom), like 3/4. An algebraic fraction is the same idea, but it includes variables, like (x + 2) / (x - 1). Just like with numerical fractions, we can add, subtract, multiply, and divide algebraic fractions, but we have to follow special rules to keep the expressions valid.

Learning how to simplify and manipulate algebraic fractions is important because it helps solve equations in algebra, makes complex problems easier to work with, and is useful in subjects like physics, engineering, and economics.

Equations

Several different types of equations are found in Calculus questions. Polynomial equations, such as quadratic and cubic equations, often appear when solving for stationary points. Rational equations, involving fractions with polynomials, arise in limits and integration. Exponential and logarithmic equations are used in growth and decay problems. Trigonometric equations appear in applications of derivatives and integrals involving periodic functions, such as motion and waves.

In more advanced topics, differential equations are introduced to model real-world changes, such as population growth and motion under gravity. Solving these equations often requires algebraic manipulation, factorisation, and substitution techniques. The ability to handle these different types of equations is essential for understanding and applying calculus concepts in various mathematical and scientific contexts.