Roots of a Polynomial - MathsTips.com

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: $P_{n}(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}$ , where $a_{n}$ is not equal to zero and n refers to the degree of a polynomial and $a_{0}, a_{1},.... a_{n}$ are real coefficient. Thus, the degree of the polynomial gives the idea of the number of roots of that polynomial. The roots may be different.

Example 1: Find the roots of the polynomial equation: $x^{2}+4x+4$

Solution: Given polynomial equation $x^{2}+4x+4$

By factoring the quadratic: $x^{2}+4x+4$ = $x^{2}+2x+2x+4 = 0$

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: $2x^{3}+7x^{2}+3$

Solution: Given polynomial equation $2x^{3}+7x^{2}+3$

By factoring the quadratic: $2x^{3}+7x^{2}+3$ = $x(2x^{2}+6x+x+3) = 0$

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 = $\frac{-1}{2}$ ,x = -3. Zeroes of polynomial are $\frac{-1}{2}$ ,-3,0.

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 $ax^{2}+bx+c=0$ , where $a\neq 0$ and $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

If the coefficients a, b, c are real, it follows that: if $b^{2}-4ac > 0$ = the roots are real and unequal, if $b^{2}-4ac = 0$ = the roots are real and equal, if $b^{2}-4ac < 0$ the roots are imaginary.

Example 1: Find the roots of the quadratic polynomial equation: $x^{2}-10x+26 = 0$

Solution: Given quadratic polynomial equation $x^{2}-10x+26 = 0$

So, a = 1,b = -10 and c = 26

By putting the formula as D = $b^{2}-4ac$ = 100 – 4 * 1 * 26 = 100 – 104 = -4 < 0

Therefore D < 0,so roots are complex or imaginary.

Now finding the value of x, using quadratic formula = $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ = $\frac{-(-10) \pm \sqrt{-4}}{2*1}$ = $\frac{10\pm 2\sqrt{-1}}{2}$ = $\frac{10\pm 2{i}}{2}$ = $5 \pm i$