Question

Find a quadratic polynomial whose sum and product of zero’s is 5 and 6 respectively.

Answer 1

Hint: Use the direct formula to get a quadratic equation when its sum of the roots and product of the roots are given, that is \[{{x}^{2}}\] - (sum of zeros) x + (product of zeros)=0.

Complete step-by-step answer:

A quadratic polynomial is a polynomial of degree 2 or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
We have been asked to find a quadratic polynomial.
We have been given the sum of zero’s as 5.
Product of zero’s as 6.
The formula for a quadratic polynomial is given by,
\[{{x}^{2}}\] - (sum of zeros) x + (product of zeros)
Thus by substituting these values we can get our quadratic polynomial as \[{{x}^{2}}-5x+6=0\].


Verification (optional)

We have been told that the sum of zeros is 5.
\[\therefore \alpha +\beta =5\]
Similarly, we have been told that the product of zeros is 6.
\[\therefore \alpha .\beta =6\]
If we take \[\alpha =2\] and \[\beta =3\] then,
\[\therefore \] Sum of zeros \[=\alpha +\beta =2+3=5\].
Product of zeros \[=\alpha \beta =2\times 3=6\].
Thus when \[\alpha =2\] and \[\beta =3\] the sum and product of zero’s are 5 and 6.
Thus we can split the \[{{2}^{nd}}\] term of a quadratic polynomial \[{{x}^{2}}-5x+6=0\].
\[\begin{align}
  & \therefore {{x}^{2}}-5x+6=0 \\
 & {{x}^{2}}-\left( 2x+3x \right)+6=0 \\
 & {{x}^{2}}-2x-3x+6=0 \\
 & x\left( x-2 \right)-3\left( x-2 \right)=0 \\
 & \therefore \left( x-2 \right)\left( x-3 \right)=0 \\
\end{align}\]
\[\therefore x-2=0\] and \[x-3=0\Rightarrow x=2\] and \[x=3\].
Thus we got the roots of the quadratic polynomial \[{{x}^{2}}-5x+6=0\] as \[x=2\] and \[x=3\].
\[\therefore \] Quadratic polynomial = \[{{x}^{2}}-5x+6=0\].

Note: In case if you don’t remember the direct formula you can assume the two roots and make two equations using the information given in the question , solve to find those two roots and write the quadratic equation as $(x-x_1)(x-x_2)=0$