Question

Find a quadratic polynomial whose zeroes are 1 and -3. Verify the relation between the coefficients and zeroes of the polynomial.

Answer 1

Hint: Assume the given zeroes of the quadratic polynomial as ‘a’ and ‘b’. Write x = a and x = b as the roots of the polynomial. In the next step find the factors of the polynomial by taking ‘a’ and ‘b’ to the L.H.S. Now, multiply, these two factors to get the required quadratic polynomial. Use the formulas: - sum of roots = \[-\dfrac{B}{A}\] and product of roots = \[\dfrac{C}{A}\] to verify the relations between the coefficients and zeroes. Here, A, B and C are the coefficients of \[{{x}^{2}}\], x, and constant respectively.

Complete step by step answer:
Here, we have been provided with zeroes of a quadratic polynomial and we have to determine that polynomial and verify the relation between the coefficients and zeroes.
Now, we know that zeroes of a polynomial are also called its roots and are defined as the values of x for which the values of polynomials becomes 0. So, we have x = 1 and x = -3 as the roots of the quadratic polynomial. We know that if x = a and x = b are the roots of a polynomial then (x – a) and (x – b) are known as the factors of a polynomial. So, we have (x – 1) and (x + 3) as the factors of the polynomial.
We know that a polynomial is a product of all its factors, so we have,
\[\Rightarrow \] f (x) = required quadratic polynomial
\[\begin{align}
  & \Rightarrow f\left( x \right)=\left( x-1 \right)\left( x+3 \right) \\
 & \Rightarrow f\left( x \right)={{x}^{2}}+3x-x-3 \\
 & \Rightarrow f\left( x \right)={{x}^{2}}+2x-3 \\
\end{align}\]
Hence, \[f\left( x \right)={{x}^{2}}+2x-3\] is our quadratic polynomial.
Now, let us verify between the coefficients and zeroes of the polynomial. We have two relations for a quadratic polynomial given as: -
(1) Sum of roots = \[-\dfrac{B}{A}\]
(2) Product of roots = \[\dfrac{C}{A}\]
Here, A, B and C are the coefficients of \[{{x}^{2}}\], x and constant term respectively in the quadratic polynomial. So, we have,
\[\Rightarrow f\left( x \right)={{x}^{2}}+2x-3\]
\[\Rightarrow \] A = 1, B = 2, C = -3
Therefore, for relation (1), we have,
\[\Rightarrow \] Sum of roots = \[-\dfrac{B}{A}\]
\[\begin{align}
  & \Rightarrow 1+\left( -3 \right)=\dfrac{-\left( 2 \right)}{1} \\
 & \Rightarrow -2=-2 \\
\end{align}\]
\[\Rightarrow \] L.H.S. = R.H.S.
Hence, verified
\[\Rightarrow \] Product of roots = \[\dfrac{C}{A}\]
\[\begin{align}
  & \Rightarrow 1\times \left( -3 \right)=\dfrac{-3}{1} \\
 & \Rightarrow -3=-3 \\
\end{align}\]
\[\Rightarrow \] L.H.S. = R.H.S.
Hence, verified
Therefore, the relations between the coefficients and zeroes of the polynomial are verified.

Note:
One may note that the difference between a quadratic polynomial and a quadratic equation. If we will substitute the obtained polynomial f (x) equal to 0 then it will be called an equation. We use this substitution to find the roots of the polynomial. You must remember the formulas of the sum of roots and the product of roots otherwise you will not be able to verify the relations (1) and (2).