Polynomial Equations – Definition, Formula and Solving Polynomial Equations

Polynomial Equations in Detail

A polynomial equation is an equation in which one or more terms are polynomials. A polynomial is an expression consisting of a finite number of terms, each of which is a product of a constant and one or more variables raised to a positive integer power. Polynomial Equations – Definition Formula and Solving Polynomial Equations.

A polynomial equation can be solved by factoring the polynomial and solving the resulting equations.

What is a Polynomial Equation?

A polynomial equation is an equation in which at least one variable appears as a power of a nonzero constant. A polynomial equation always has a finite number of solutions.

Degree of Polynomial

A polynomial is a mathematical expression consisting of one or more terms, each of which is an algebraic sum of a fixed number of monomials. A degree of a polynomial is the sum of the exponents of the variables in the polynomial.

Polynomial Formula

A polynomial is an expression that is made up of one or more terms, each of which is a multiple of a power of x. The polynomial expression is written in descending order from the highest power of x to the lowest.

The degree of a polynomial is the highest power of x in the expression. The terms of a polynomial are separated by plus or minus signs, and parentheses may be used to group terms.

A polynomial can be factored into a product of polynomials, each of which has a lower degree than the original polynomial.

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

Solving Polynomial Equations

A polynomial equation is an equation that contains a polynomial. A polynomial is an expression that contains one or more terms, each of which is a product of a constant and a variable raised to a nonnegative integer power.

A polynomial equation can be solved using the Quadratic Formula.

Application of Polynomial Equations in Real Life

Polynomial equations can be used in a number of different ways in the real world. One way is to find the curve of a graph that best fits the data points. This is often done with polynomial regression. Polynomial equations can also be used to solve problems in physics and engineering. For example, they can be used to find the displacement of an object at a certain time or the force on an object at a certain point in space.

Polynomial Equations – Definition Formula and Solving Polynomial Equations.