Polynomials (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Key terminology

Understanding polynomial terminology is essential for working with these mathematical expressions effectively. Let's explore the fundamental concepts you need to know.

An expression is a term or group of terms that consists of numbers, variables and the basic operators (+, -, ×, ÷). These form the building blocks of algebraic mathematics and allow us to represent mathematical relationships in a structured way.

A univariate expression contains only one variable, making it simpler to analyse and solve. This is in contrast to multivariate expressions which involve two or more variables.

The root or zero of an equation refers to the value of x such that $f(x) = 0$ is satisfied. Finding roots is crucial for solving equations and understanding where graphs intersect the x-axis.

Types of polynomials

Polynomials are classified based on the number of terms they contain, which helps us identify appropriate solution methods.

A polynomial is an expression that involves one or more variables having different powers and coefficients. The general form is written as $a_nx^n + ... + a_2x^2 + a_1x + a_0$, where $n \in \mathbb{N}_0$. The coefficients $(a_0, a_1, a_2, \text{etc.})$ can be any real numbers, and the exponents must be non-negative integers.

A monomial is a polynomial with exactly one term. Examples include $7a^2b$ or $15xy^2$. These are the simplest polynomial expressions and often serve as building blocks for more complex polynomials.

A binomial contains exactly two terms. Common examples are $2x + 5z$ or $26 - y^2k$. Binomials frequently appear in factoring problems and algebraic expansions.

A trinomial has exactly three terms. You'll often encounter these in quadratic expressions like $a - b + c$ or $4x^2 + 17xy - y^3$. Trinomials are particularly important when working with quadratic equations.

Degree and order

The degree (also called the order) of a univariate polynomial is the value of the highest exponent in the polynomial. For example, in the expression $7p - 12p^5 + 3p^7 + 8$, the highest exponent is 7, so the degree is 7. The degree tells us important information about the polynomial's behaviour and the maximum number of roots it can have.

Essential formulas and theorems

Several key mathematical tools help us work with polynomials effectively.

Quadratic formula

The quadratic formula provides a direct method for solving quadratic equations:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula works for any quadratic equation in the form $ax^2 + bx + c = 0$, where $a \neq 0$. The discriminant $(b^2 - 4ac)$ tells us about the nature of the roots.

Remainder theorem

The remainder theorem states that when a polynomial $p(x)$ is divided by $cx - d$, the remainder equals $p(d/c)$. This theorem provides a quick way to find remainders without performing long division.

Factor theorem

The factor theorem builds on the remainder theorem. If a polynomial $p(x)$ is divided by $cx - d$ and the remainder $p(d/c)$ equals zero, then $cx - d$ is a factor of $p(x)$. This theorem is invaluable for factoring polynomials and finding their zeros.

Converse of the factor theorem

The converse of the factor theorem states that if $cx - d$ is a factor of $p(x)$, then $p(d/c) = 0$. This relationship helps us verify factors and find additional roots of polynomial equations.

Synthetic division

Synthetic division provides an efficient method for dividing polynomials, particularly when dividing by linear factors.

The process involves setting up a systematic arrangement where we determine the coefficients of the quotient through calculation. We use the relationships:

• $q_2 = a_3 + (q_3 \times d/c)$, where $q_3 = 0$ for the highest term
• $q_1 = a_2 + (q_2 \times d/c)$
• $q_0 = a_1 + (q_1 \times d/c)$
• $R = a_0 + (q_0 \times d/c)$