Question

How many zeros does a cubic polynomial have?

Answer 1

Hint: For a polynomial, there may be few (one or more) values of the variable for which the polynomial may result in zero. These values are known as zeros of a polynomial. We can say that the zeros of a polynomial are defined as the points where the polynomial equals to zero on the whole.
Complete step by step answer:
A polynomial which touches the axis thrice will be a cubic polynomial. Suppose that the polynomial is in x and the highest power of the polynomial is 3. Therefore The graph of that polynomial will touch the x-axis thrice, the point at which it will touch the x-axis will be denoted as \[\left( {a,0} \right)\] Now let us say that any cubic polynomial is of the form
\[f\left( x \right) = a{x^2} + b{x^2} + cx + d\] Where we can write \[f\left( x \right) = y\] Now it can be clearly observed that in the points \[\left( {a,0} \right)\] when the graph touches the x-axis at a the value of \[f\left( x \right)\] i.e., y automatically becomes 0. Therefore it will have 3 zeros or roots.

Note: In any polynomial The highest power of the variable will be equal to the number of zeros or roots it will have, the roots can all be equal like for cubic polynomial \[f\left( x \right) = {(x - 1)^3}\] here all the roots of this polynomial will be 1 only. Also note that there may be roots existing in a complex plane but it will exist and the complex roots always exist in pairs that means for polynomials you will always get at least 1 real root.