Question

How do you know how many roots are in a polynomial?

Answer 1

Hint: To know the number of roots of a polynomial we need to analyze the discriminant of the polynomial if the given polynomial is quadratic or cubic in nature. Also to know the number of roots of a polynomial we will analyze the fundamental theorem of algebra.

Complete step by step answer:
We know that polynomial is an expression that has many terms. A polynomial has variables, constants and the exponents. For example $5{{x}^{2}}+2x+3$ is a polynomial where $5,2,3$ are constants and x is a variable.
We know that the roots of a polynomial are those values of a variable that cause the polynomial to evaluate to zero. Roots of a polynomial are solutions for a given polynomial also known as zeros of the polynomial.
According to the fundamental theorem of algebra a polynomial has exactly as many roots as its degree. Degree is the highest exponent of the polynomial. For example a polynomial with degree 2 has 2 roots.
Let us understand this with an example.
Let us consider a quadratic equation $5{{x}^{2}}+13x-6$.
Now, we know that the quadratic formula to find the roots of quadratic equation is given as
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values in the above formula we get
$\Rightarrow x=\dfrac{-13\pm \sqrt{{{13}^{2}}-4\times 5\times -6}}{2\times 5}$
Now, on solving the obtained equation we get
$\begin{align}
  & \Rightarrow x=\dfrac{-13\pm \sqrt{169+120}}{10} \\
 & \Rightarrow x=\dfrac{-13\pm \sqrt{289}}{10} \\
\end{align}$
$\Rightarrow x=\dfrac{-13\pm 17}{10}$
Now, we know that a quadratic equation has two roots. We can write the obtained equation as
$\Rightarrow x=\dfrac{-13+17}{10}and\Rightarrow x=\dfrac{-13-17}{10}$
Now, let us first consider
 $\Rightarrow x=\dfrac{-13+17}{10}$
On solving we get
$\begin{align}
  & \Rightarrow x=\dfrac{4}{10} \\
 & \Rightarrow x=\dfrac{2}{5} \\
\end{align}$
Now, let us consider $\Rightarrow x=\dfrac{-13-17}{10}$
On solving we get
$\begin{align}
  & \Rightarrow x=\dfrac{-30}{10} \\
 & \Rightarrow x=-3 \\
\end{align}$
So the two factors of the equation $5{{x}^{2}}+13x-6$ will be $\left( 5x-2 \right)\left( x+3 \right)$.

Hence we can say that a polynomial have exactly as many roots as its degree.

Note: The discriminant of a cubic polynomial is given by the formula $D={{b}^{2}}{{c}^{2}}-4a{{c}^{3}}-4{{b}^{3}}d-27{{a}^{2}}{{d}^{2}}+18abcd$. The value of the discriminant tells the nature of roots of a polynomial whether the roots are distinct real roots or complex roots.