Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
\[0,\sqrt{5}\]

Answer 1

Hint: First of all, try to recall the relationship between the coefficients and zeros of the quadratic polynomial. Now, we can write the quadratic polynomial as \[{{x}^{2}}\]– (Sum of the root)x + (Product of roots) = 0 and can verify it by using \[\alpha +\beta =\dfrac{-b}{a}\] and \[\alpha \beta =\dfrac{c}{a}\].

Complete step-by-step solution -
Here, we are given 0 and \[\sqrt{5}\] and we have to find the quadratic polynomial in terms of the sum or the product of its zeroes. Before proceeding with the question, let us know what quadratic polynomial is. A quadratic polynomial or quadratic equation is a polynomial of degree 2 and it has two roots. The general form of the quadratic equation is \[a{{x}^{2}}+bx+c=0\]. It has a maximum of 2 zeroes which can be real or imaginary.
If \[\alpha \] and \[\beta \] are the roots of the quadratic equation, then we can write \[\alpha +\beta =\dfrac{-b}{a}\] and \[\alpha \beta =\dfrac{c}{a}\].
Also, if we have roots as \[\alpha \] and \[\beta \], then we can write the quadratic equation as
\[{{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0\]
\[{{x}^{2}}\]– (Sum of root)x + (Product of roots) = 0
Now, let us consider our question. Here, we are given two zeroes of the polynomial is 0 and \[\sqrt{5}\]. Let the two zeroes of the quadratic polynomial are \[\alpha \] and \[\beta \] where \[\alpha =0\] and \[\beta =\sqrt{5}\].
Then, we get the sum of zeroes of the quadratic equation \[=\alpha +\beta \]
\[=0+\sqrt{5}\]
\[=\sqrt{5}.....\left( i \right)\]
Also, we get the product of zeroes of the quadratic equation \[=\alpha \beta \]
\[=0\times \sqrt{5}\]
\[=0....\left( ii \right)\]
We know that we can write any quadratic equation as
\[{{x}^{2}}\]– (Sum of root)x + (Product of roots) = 0
\[\Rightarrow {{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0\]
By substituting the values of \[\left( \alpha +\beta \right)\] and \[\alpha \beta \] from equation (i) and (ii), in the above equation, we get,
\[{{x}^{2}}-\left( \sqrt{5} \right)x+0=0\]
\[\Rightarrow {{x}^{2}}-\sqrt{5}x=0\]
Now, we can verify the relation between coefficient and sum and product of the roots. By comparing the above equation by \[a{{x}^{2}}+bx+c=0\].
We get,
a = 1
\[b=-\sqrt{5}\]
c = 0
We know that,
\[\alpha +\beta =\dfrac{-b}{a}\]
So, we get,
\[\left( \sqrt{5} \right)=-\dfrac{\left( -\sqrt{5} \right)}{1}\]
\[\sqrt{5}=\sqrt{5}\]
LHS = RHS
We also know that,
\[\alpha \beta =\dfrac{c}{a}\]
\[\Rightarrow 0=\dfrac{0}{1}\]
0 = 0
LHS = RHS
So, we have verified the relationship between zeroes and the coefficient of a quadratic polynomial.

Note: In this question, students can cross-check their answer by further splitting the final quadratic polynomial into two factors and checking if the roots are the same as given initially or not. We can do this as follows:
Our quadratic polynomial is:
\[{{x}^{2}}-\sqrt{5}x=0\]
By taking out x common, we get,
\[x\left( x-\sqrt{5} \right)=0\]
So, we get, x = 0 and \[x=\sqrt{5}\].
Hence, our answer is correct.