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  • For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation minus 3

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation minus 3

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Answer.         \left [\frac{\sqrt{5}}{2}, \frac{-1}{\sqrt{5}} \right ]

Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.

Here sum of zeroes \frac{-3}{2\sqrt{5}}

Product of zeroes =-\frac{1}{2}

Let p(x) is the required polynomial

p(x) = x2 – (sum of zeroes)x + (product of zeros)

p(x)=x^{2}-\left (-\frac{-3}{2\sqrt{5}} \right )x+ \frac{-1}{2}

p(x)=x^{2}+\frac{-3}{2\sqrt{5}} x- \frac{1}{2}

Multiplying by 2 \sqrt{5}  we get

p(x)=2 \sqrt{5}x^{2}+3x-\sqrt{5}

Hence, 2 \sqrt{5}x^{2}+3x-\sqrt{5}  is the required polynomial

p(x)=2 \sqrt{5}x^{2}+3x-\sqrt{5}

=2 \sqrt{5}x^{2}-5x+2x -\sqrt{5}

\sqrt{5}(2x-\sqrt{5})+1(2x-\sqrt{5})

(\sqrt{5}x +1)(2x-\sqrt{5})

\left [\frac{\sqrt{5}}{2}, \frac{-1}{\sqrt{5}} \right ]  are the zeroes of p(x).

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