Degree of an Equation

The degree of an algebraic equation is defined as the degree of the polynomial $A(x)$ once the equation has been rewritten in its standard form, that is, $ A(x) = 0 $, where $A(x)$ is a polynomial in $x$.

In other words, the degree of the equation is determined by the highest power of the variable $x$ appearing in the polynomial.

Why is knowing the degree important?

The degree of an equation is crucial because it tells us how many solutions the equation can have (at most as many as its degree) and guides us in choosing the most appropriate method for solving it.

Identifying the degree of an equation is therefore the first step toward selecting the right strategy for finding its solutions.

A Practical Example

Let’s consider the following equation:

$$ 2x^2 + 2x - 3 = 5x + 2 $$

Any algebraic equation can be rewritten so that all terms appear on the same side of the equals sign.

This is done using the transposition rule, which allows us to move terms from one side of the equation to the other, changing their sign in the process.

Applying the transposition rule, we move $5x + 2$ to the left-hand side:

$$ 2x^2 + 2x - 3 - 5x - 2 = 0 $$

Combining like terms and arranging them in descending powers of $x$, we arrive at the standard form (or canonical form) of the equation: $$ A(x) = 0 $$

$$ 2x^2 - 3x - 5 = 0 $$

The equation is now in standard form.

The polynomial $ A(x) = 2x^2 - 3x - 5 $ is of degree two because the highest exponent of $ x $ is $2$.

Therefore, the equation $ 2x^2 - 3x - 5 = 0 $ is a quadratic equation, i.e. an equation of degree two.

Example 2

Consider the equation:

$$ 3x^4 - 7x^2 + x - 10 = 0 $$

Here, the polynomial $A(x)$ is $3x^4 - 7x^2 + x - 10$, and the highest power of $x$ is $4$.

Therefore, this is a fourth-degree equation.

Example 3

Let’s look at the following equation:

$$ x^2 + 2x - 3 = x^2 - 2 $$

At first glance, it might appear to be a quadratic equation. However, that’s incorrect because the equation hasn’t yet been transformed into standard form.

We move all terms to the left-hand side:

$$ x^2 + 2x - 3 - x^2 + 2 = 0 $$

Then simplify:

$$ 2x - 1 = 0 $$

Now the equation is in standard form, and as we can see, it’s actually a linear equation - that is, of degree one.

Note. This example is particularly important because it highlights that the degree of an equation can only be determined once the equation has been expressed in its standard form $ A(x)=0 $. Until then, appearances can be misleading.

Examples of Equations of Different Degrees

Here are a few more examples to further clarify the concept:

• First-Degree Equation
  This is a first-degree equation because the highest power of $x$ is 1. $$ 5x - 8 = 0 $$ A first-degree equation can be solved simply by isolating $x$: $$ x = \frac{8}{5} $$ These are the simplest equations to solve. First-degree equations are also known as “linear equations.”
• Second-Degree Equation
  This is a second-degree equation because the highest power of $x$ is 2. $$ x^2 - 4x + 3 = 0 $$ A second-degree equation is solved using the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ or by completing the square. You can read more about quadratic equations.
• Third-Degree Equation
  This is a cubic equation, or equation of degree three: $$ x^3 - 2x^2 + x - 1 = 0 $$ There are specific techniques for solving cubic equations, such as factoring out the variable and applying the zero-product property. $$ x \cdot \left( x^2 - 2x + 1 - \frac{1}{x} \right) = 0 $$ For more information on solving higher-degree equations.

And so on.