A polynomial is defined as the sum of more than one or more algebraic terms where each term consists of several degrees of same variables and integer coefficient to that variables. x2−3×2−3, 5×4−3×2+x−45×4−3×2+x−4 are some examples of polynomials. The roots or also called as zeroes of a polynomial P(x) for the value of x for which polynomial P(x) is equal to 0. In other words, we can say that polynomial P(x) will have the same value of x if x=r i.e. the value of the root of the polynomial that will satisfy the equation P(x) = 0. These are sometimes called solving the polynomial. The degree of the polynomial is always equal to the number of roots of polynomial P(x).
Definition
In any polynomial, the root is that the value of the variable that satisfies the polynomial. Polynomial is an expression consisting of variables and coefficients of the form:
Example 1: Find the roots of the polynomial equation:
Solution: Given polynomial equation
By factoring the quadratic:
x(x+2) + 2(x+2) = 0 therefore, (x+2)(x+2)=0
Set each factor equal to zero: x+2 =0 or x+2 = 0
So, x=-2 or x=-2 . Both the roots are same, i.e. -2.
Example 2: Find the roots of the polynomial equation:
Solution: Given polynomial equation
By factoring the quadratic:
x(2x(x + 3) + (x + 3)) = 0 therefore, x(2x + 1)(x + 3) = 0
Set each factor equal to 0: x = 0,2x+1 = 0,x+3 = 0
So, x = 0,x =
Quadratic roots of Polynomial
Roots are the solution to the polynomial. The roots may be real or complex (imaginary), and they might not be distinct. A quadratic equation is
If the coefficients a, b, c are real, it follows that: if
Example 1: Find the roots of the quadratic polynomial equation:
Solution: Given quadratic polynomial equation
So, a = 1,b = -10 and c = 26
By putting the formula as D =
Therefore D < 0,so roots are complex or imaginary.
Now finding the value of x, using quadratic formula =
Therefore, the roots are 5 + i and 5 – i.
Example 2: Find the roots of the quadratic polynomial equation:
Solution: Given polynomial equation:
So, roots are
Therefore, x =
Exercise
Find the roots of polynomials by factoring:
Find the roots of the quadratic polynomial equation: