Types of Polynomial Equation
A polynomial equation is basically of four types;
- Monomial Equations
- Binomial Equations
- Trinomial or Cubic Equations
- Linear Polynomial Equations
- Quadratic Polynomial Equations
- 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