Question

If the product of two zeros of the polynomial equation $p(x)={{x}^{3}}-6{{x}^{2}}+11x-6$ is 3 then what is the value of its third zero?

Answer 1

In the above question it is mentioned that the product of two zeros is 3 and we need to calculate the third zero in this zero signifies the roots of the polynomial equation and we know the two roots and need to calculate the third roots. There are some properties of polynomial roots through which we will be easily able to calculate the third root. The property to be used is of the product of roots.

Complete step by step solution:
In the above question the mentioned polynomial equation is $p(x)={{x}^{3}}-6{{x}^{2}}+11x-6$. As the highest power of x in the polynomial function is 3 there will be 3 roots to the polynomial function. From the property of the polynomial equation which states that the product of all the roots of the equation is equal to $-\dfrac{d}{a}$. Let us take the roots of the given polynomial equation as u, v, w. so from the product of roots property of the equation it can also be said as:
$uvw=-\dfrac{d}{a}......\left( 1 \right)$
Where a is the coefficient of ${{x}^{3}}$ and d is the constant term from the polynomial equation we can see that the value of a is 1 and the value of d is -6. Now by substituting the value of a and d which we got from the polynomial equation in equation 1 we get,

& \Rightarrow uvw=-\dfrac{\left( -6 \right)}{1} \\\ & \Rightarrow uvw=6......\left( 2 \right) \\\ \end{aligned}$$ Now it is also mentioned that the value of product of two roots is equal to 3 so we can also say that $$uv=3$$ and now we will substitute in equation 2 and we get, $$\begin{aligned} & \Rightarrow 3w=6 \\\ & \Rightarrow w=2 \\\ \end{aligned}$$ From the above calculation we can see that the value of w is equal to 2. **The value of the third root of the polynomial equation is $$w=2$$** **Note:** In the above type of questions it will be better to learn all the properties of the polynomial equation to make the calculation easier. Not only for this type of questions but also to directly find out the sum of roots and the product of roots (which we used to find the value of third root in this question)