Polynomial Equation

A polynomial equation is an equation that contains at least one algebraic term with one or more variables, where all exponents are non-negative integers. The highest exponent determines the degree of the equation, and each part of the equation is called a term. 

For example, 

x² + 3x – 1 is a polynomial, and x² + 3x – 1 = 0 is a polynomial equation.

In general, a polynomial equation is always of the form: 

Polynomial function = 0

Here are a few examples of polynomial equations:

1. 5x⁵ – x² + 3x + 2 = 0
2. x⁴ – 4x³ + x – 1 = 0
3. 2x³ + 5x – 8 = 0

However, the following are not polynomial equations:

• ${x^{2}+x^{\dfrac{3}{2}}-1=0}$ 
• ${\sqrt{2}x^{3}+x-1=0}$

Formula

Mathematically, a polynomial equation is written as a_n(xⁿ), which, on expanding using the Binomial theorem, we get

P(x) = a_nxⁿ + a_{n – 1}x^{n – 1} + … + a₁x + a₀ = 0

Here,

• x is variable
• P(x) is polynomial with variable x
• a_n, a_{n – 1}, …, a₁, a₀ are coefficients, with ∀a_n ∈ ℝ
• n is a non-negative integer, which represents the degree of P(x)

Types

Polynomial equations are classified into four main types based on their degree.

Linear Equation

A polynomial equation of degree 1 is called a linear equation. In such equations, each term is either a constant, a variable, or a product of a constant and a single variable. Linear equations represent straight lines when graphed on a coordinate plane. 

It can be expressed in the standard form as: 

ax + b = 0

Here, a and b are constants, and a ≠ 0

For example,

• 2x + 3 = 0
• x = 0

Quadratic Equation

A polynomial equation of degree 2 is called a quadratic equation. These equations represent parabolas when graphed on a coordinate plane, which can open upwards or downwards depending on the coefficients. 

The standard form of a quadratic equation is: 

ax² + bx + c = 0

Here, a, b, and c are constants, and a ≠ 0

For example,

• 2x² + 2x + 3 = 0
• x² + 1 = 0

The height of a projectile in motion can be described by a quadratic polynomial equation.

Cubic Equation

A polynomial equation of degree 3 is called a cubic equation. Cubic equations represent an S-shaped curve when graphed on a coordinate plane, often changing direction once or twice.

In standard form, it is written as:

ax³ + bx² + cx + d = 0

Here, a, b, c, and d are constants, and a ≠ 0

For example,

• x³ – 5x² + x – 6 = 0
• 2x³ + 5 = 0

Solving (Finding Roots)

To solve a polynomial equation, we find the values of x that satisfy the equation. These solutions, known as the roots of the polynomial, are found by setting the equation equal to zero and solving for the variable.

Let us find the roots of the polynomial equation 5x + 2 = 4

Now, solving the equation, 

5x + 2 = 4

⇒ 5x = 4 – 2

⇒ 5x = 2

⇒ x = ${\dfrac{2}{5}}$

Thus, x = ${\dfrac{2}{5}}$ is the root of 5x + 2 = 4

Quadratic Polynomial

There are two methods commonly used to find the roots of quadratic polynomial equations.

By Factoring Method

In this method, the quadratic equation must be factored with zero isolated on one side. We first write the quadratic equation in standard form, ax² + bx + c = 0, and then factor the expression on the left.

Let us find the roots of -5x² + 20x – 15 = 0

Now, solving the equation,

-5x² + 20x – 15 = 0

⇒ x² – 4x + 3 = 0 (dividing by -5)

⇒ x² – 3x – x + 3 = 0 (breaking down the middle term)

⇒ x(x – 3) – 1(x – 3) = 0 (taking out the common factors)

⇒ (x – 3)(x – 1) = 0

Using zero product property,

(x – 3) = 0 and (x – 1) = 0

⇒ x = 3 and x = 1

Thus, x = 1 and 3 are the roots of the polynomial equation -5x² + 20x – 15 = 0

By Quadratic Formula

The roots of a quadratic equation whose degree is 2, such as ax² + bx + c = 0, are evaluated using the formula 

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

Let us find the roots of 3x² – 5x – 2 = 0

Comparing the equation with ax² + bx + c = 0, we get

a = 3, b = -5, and c = -2

Using the quadratic formula, 

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

⇒ x = ${\dfrac{-\left( -5\right) \pm \sqrt{\left( -5\right) ^{2}-4\times 3\times \left( -2\right) }}{2\times 3}}$

⇒ x = ${\dfrac{5\pm \sqrt{25+24}}{6}}$

⇒ x = ${\dfrac{5\pm \sqrt{49}}{6}}$

⇒ x = ${\dfrac{5\pm 7}{6}}$

⇒ x = 2 and ${-\dfrac{1}{3}}$

Thus, x = 2 and ${-\dfrac{1}{3}}$ are the roots of 3x² – 5x – 2 = 0

Cubic Polynomial

To find the roots of a cubic polynomial (of degree 3), the polynomial equation is first factorized to obtain both a linear and a quadratic equation. Then, we determine the zeros of the polynomial.

Let us find the roots of the polynomial x³ – 6x² + 11x – 6 = 0

⇒ x³ – x² – 5x² + 5x + 6x – 6 = 0

⇒ x²(x – 1) – 5x(x – 1) + 6(x – 1) = 0

⇒ (x – 1)(x² – 5x + 6) = 0

Using the zero product property,

(x – 1) = 0 and (x² – 5x + 6) = 0

Here, (x – 1) = 0

⇒ x = 1 …..(i)

Also, (x² – 5x + 6) = 0

⇒ x² – 3x – 2x + 6 = 0

⇒ x(x – 3) – 2(x – 3) = 0

⇒ (x – 3)(x – 2) = 0

Again, using the zero product property,

(x – 3) = 0 and (x – 2) = 0

⇒ x = 3 and 2 …..(ii)

Thus, from (i) and (ii), x = 1, 2, and 3 are the roots of x³ – 6x² + 11x – 6 = 0