If $\alpha ,\ \beta$ are the zeroes of a polynomial, such that $\alpha+\beta=6$ and $\alpha\beta=4$, then write the polynomial.
Given: $\alpha$ and $\beta$ are the zeroes of a polynomial. $\alpha+\beta=6$ and $\alpha\beta=4$
What to do: To write the polynomial.
Solution:
Since it has two zeroes, it is a quadratic polynomial.
And its zeroes are $\alpha$ and $\beta$.
$\Rightarrow ( x-\alpha ) =0\ and\ ( x-\beta ) =0$
$\Rightarrow ( x-\alpha )( x-\beta ) =0$
$\Rightarrow x^{2} -\alpha x-\beta x+\alpha\beta=0$
$\Rightarrow x^{2} -( \alpha+\beta) x+\alpha\beta =0$
$\therefore$The polynomial is $x^{2} -( \alpha+\beta) x+\alpha\beta =0$
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