Polynomial Equations

Polynomial equations are one of the significant concepts of Mathematics, where the relation between numbers and variables are explained in a pattern. In Maths, we have studied a variety of equations formed with algebraic expressions. When we talk about polynomials, it is also a form of the algebraic equation.

What is a Polynomial Equation?

The equations formed with variables, exponents and coefficients are called as polynomial equations. It can have different exponents, where the higher one is called the degree of the equation. We can solve polynomials by factoring them in terms of degree and variables present in the equation.

A polynomial function is an expression which consists of a single independent variable, where the variable can occur in the equation more than one time with different degree of the exponent. Students will also learn here how to solve these polynomial functions. The graph of a polynomial function can also be drawn using turning points, intercepts, end behaviour and the Intermediate Value Theorem.

Example of polynomial function:

f(x) = 3x² + 5x + 19

Read More: Polynomial Functions

Polynomial Equations Formula

Usually, the polynomial equation is expressed in the form of a_n(xⁿ). Here a is the coefficient, x is the variable and n is the exponent. As we have already discussed in the introduction part, the value of exponent should always be a positive integer.

If we expand the polynomial equation we get;

F(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + …….. + a₁x +a₀ = 0

This is the general expression and it can also be expressed as;

F(x) = \(\sum_{k=0}^{n}a_{k}n^k = 0\)

Example of a polynomial equation is: 2x² + 3x + 1 = 0, where 2x² + 3x + 1 is basically a polynomial expression which has been set equal to zero, to form a polynomial equation.

Types of Polynomial Equation

A polynomial equation is basically of four types;

1. Monomial Equations
2. Binomial Equations
3. Trinomial or Cubic Equations
4. Linear Polynomial Equations
5. Quadratic Polynomial Equations
6. Cubic Polynomial Equation

Monomial Equation:

An equation which has only one variable term is called a Monomial equation. This is also called a linear equation. It can be expressed in the algebraic form of;

ax + b = 0

For Example:

• 4x + 1 = 0
• 5y = 2
• 8z – 3 = 0

Binomial Equations:

An equation which has only two variable terms and is followed by one variable term is called a Binomial equation. This is also in the form of the quadratic equation. It can be expressed in the algebraic form of;

ax² + bx + c = 0

For Example:

• 2x² + 5x + 20 = 0
• 3x² – 4x + 12 = 0

Trinomial Equations:

An equation which has only three variable terms and is followed by two variable and one variable term is called a Trinomial equation. This is also called a cubic equation. In other words, a polynomial equation which has a degree of three is called a cubic polynomial equation or trinomial polynomial equation.

Since the power of the variable is the maximum up to 3, therefore, we get three values for a variable, say x.

It is expressed as;

a₀ x³ + a₁x² + a₂x + a₃ = 0, a ≠ 0

or

ax³ + bx² + cx + d = 0

For Example:

• 3x³ + 12x² – 8x – 10 = 0
• 9x³ + 5x² – 4x – 2 = 0

To get the value of x, we generally use, trial and error method, in which we start putting the value of x randomly, to get the given expression as 0. If for both sides of the polynomial equation, we get 0 ,then the value of x is considered as one of its roots. After that we can find the other two values of x.

Let us take an example:

Problem: y³ – y² + y – 1 = 0 is a cubic polynomial equation. Find the roots of it.

Solution: y³ – y² + y – 1 = 0 is the given equation.

By trial and error method, start putting the value of x.

If y = -1, then,

(-1)³ – (-1)² -1 + 1 = 0

-1 – 1 – 1 – 1 = 0

-4 ≠ 0

If y = 1, then,

1³ – 1² + 1 – 1 = 0

0 = 0

Therefore, one of the roots is 1.

y = 1

(y – 1) is one of the factors.

Now dividing the given equation with (y – 1), we get,

(y – 1) (y² +  1) = 0

Therefore, the roots are y = 1 which is a real number and y² + 1 gives complex numbers or imaginary numbers.

Quadratic Polynomial Equation

A polynomial equation which has a degree as two is called a quadratic equation. The expression for the quadratic equation is:

ax² + bx + c = 0 ; a ≠ 0

Here, a,b, and c are real numbers. The roots of quadratic equations will be two values for the variable x. These can be found by using the quadratic formula as:

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

Also Check: Polynomial Equation Solver

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