Question

Find a quadratic polynomial whose zeros are reciprocal of the zero of polynomial f(x) = ax2+bx+c

Answer 1

Let the roots of the above equation be alpha and beta. Then the roots of the new equation will be 1/alpha and 1/beta.
In the first equation...
$\alpha + \beta = - \frac{b}{a} \\ \alpha \beta = \frac{c}{a}$
If roots of an equation are given, the equation will be
${x}^{2} - ( \alpha + \beta )x + \alpha \beta$
Which in this case are the reciprocals of the given equation.
So, we will find the sum and product of the reciprocal roots.
$\frac{1}{ \alpha } + \frac{1}{ \beta } = \frac{ \alpha + \beta }{ \alpha \beta } \\ = \frac{ - \frac{b}{a} }{ \frac{c}{a} } \\ = - \frac{b}{c}$
$\frac{1}{ \alpha } \times \frac{1}{ \beta } = \frac{1}{ \alpha \beta } \\ = \frac{1}{ \frac{c}{a} } \\ = \frac{a}{c}$
Now we will put these values in the equation which can be found using roots.
${x}^{2} - ( - \frac{b}{c} )x + \frac{a}{c} = 0 \\ {x}^{2} + \frac{b}{c} + \frac{a}{c} = 0$
Multiplying the whole equation by c...
$c {x}^{2} + bx + a = 0$
Remember this equation.
If u are asked to give an equation with the roots as reciprocals of the given equation, then juat reverse the order of the coefficients.
The constant becomes the coefficient of x^2.
The coefficient of x^2 becomes the constant.
The coefficient of x remains the same.

Answer 2

Answer:

cx^2 + bx + a.

Step-by-step explanation:

Let p, q be zeros of ax^2+bx+c

therefore p+q=-$\frac{b}{a}$  and p×q=$\frac{c}{a}$

Let p and q be zeros of required polynomial

It is given that p = $\frac{1}{p}$  and q = $\frac{1}{q}$

Then,

p+q = $\frac{1}{p}$ + $\frac{1}{q}$

      = $\frac{p+q}{p*q}$

     = $\frac{\frac{-b}{a} }{\frac{c}{a} }$

    =$-\frac{b}{c}$

and  p × q =$\frac{1}{p}* \frac{1}{q}$

                =$\frac{1}{p*q}$

               = $\frac{a}{c}$

therefore,,required polynomial is:

         

                                = x^2 +$\frac{b}{c}$ x + $\frac{a}{c}$

                        I. e, = cx^2 + bx + a.

QUADRATIC POLYNOMIAL::

       

                                   Quadratic polynomial is a polynomial in which the highest degree monomial is of the second degree. A quadratic polynomial is also known as a second-order polynomial.

The project code is #SPJ2