Algebraic equation is also known as polynomial equations. A polynomial expression is made up of constants and variable, combined using mathematical operators. When a polynomial expression is equated to some constant value, it becomes polynomial equation. To understand this in simple terms, go through the following example.
Is a polynomial equation, here x is a variable, 5 and 6 are constant and mathematical operators used are addition and square, but
x²-5x+6=0 is a polynomial equation or algebraic equation because it is equated to a constant (in this case zero).
The general form of a polynomial equation is
As one can notice the power f x keep on decreasing. The above equation is called a polynomial degree of order n.
What is degree of polynomial?
Degree of a polynomial is the highest power of variable. Degree of a polynomial is always a non-negative integer.
Depending upon the degree of polynomial, polynomial may de defined as quadratic equation, Linear equation or constants.
To be very particular a polynomial equation of degree 2 is called quadratic equation, polynomial equation of degree 1 is called linear equation and polynomial equation with degree of polynomial zero are nothing but constants values.
In this module we will focus on quadratic equations
The general form of a polynomial equation is ax²+bx+c= 0
· a is the coefficient of x2
· b is the coefficient of x And
· c is the constant term
To solve quadratic equations there are three methods available, they are:
· completing the square method
· using quadratic formula
Let us see each method is detail.
In this method the quadratic equation is decomposed into 2 factors. (What are factors? Factors are those values, which if substituted into the original equation, then it result out to zero. ) for example in our example, x²-5x+6 =0 the equation can be written in the form, (x-2) (x-3)=0 . If one opens the bracket and multiply the terms then we get the original equation x²-5x+6 . Therefore (x-2) (x-3) are known as factors of the equation x²-5x+6 =0
How to find factors of any quadratic equation?
To find the factors of any quadratic equation, follow these steps:
Step 1 Decompose the middle term, i.e., the terms containing the x and it coefficient into 2 parts, such that the product of those two terms should be equivalent to the product of x2 term and the constant term. Also, the summation of those decomposed two terms should give back the original middle term.
Step 2 Take common out from the first two terms, and from second two terms in such a way, that you get common factor among both these terms.
Step 3 Again take the common factor out, and then you are left with 2 final factors of the original equation.
If the above mentioned steps are not clear to you, go through the following steps:
Let’s start with
x²-5x+6 = 0
Now let us write 5x as -2x -3x
x²-2x -3x+6 =0
This is so because, product of (-2x) and (-3x) is (+6x2), which is same as product of (x2) and (6), i.e., (6x2), the product of x2 term and constant term.
Now, take x common out from x2 and (-2x) and (-3) common out from (-3x) and (6)
The equation becomes x(x-2)-3(x-2)=0
Now take (x-2) common out
(x-2) (x-3)= 0
These are the factors of the original equation x²-5x+6=0
The values of x in this case would be 2 and 3.