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 factorization.
(i) $- \frac{8}{3}, \frac{4}{3}$ (ii) $\frac{21}{8}, \frac{5}{16}$ (iii) $- 2 \sqrt{3}, - 9$ (iv) $\frac{- 3}{2 \sqrt{5}}, - \frac{1}{2}$ Question

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Solution

We know that a quadratic polynomial whose sum and product of zeroes are given is
$f (x) = k \{x^{2} - (Sum of zeroes) x + Product of zeroes\}$

(i) We have, sum of zeroes = $\frac{- 8}{3}$ and product of zeroes = $\frac{4}{3}$
So, the required quadratic polynomial will be $f (x) = k (x^{2} + \frac{8}{3} x + \frac{4}{3})$
$f (x) = k (x^{2} + \frac{8}{3} x + \frac{4}{3}) = \frac{k}{3} (3 x^{2} + 8 x + 4) = \frac{k}{3} (3 x^{2} + 6 x + 2 x + 4) = \frac{k}{3} (3 x (x + 2) + 2 (x + 2)) = \frac{k}{3} (3 x + 2) (x + 2)$
Now, the zeroes are given by f(x) = 0.
Thus, $x = - \frac{2}{3} and x = - 2$
(ii) We have, sum of zeroes = $\frac{21}{8}$ and product of zeroes = $\frac{5}{16}$
So, the required quadratic polynomial will be $f (x) = k (x^{2} - \frac{21}{8} x + \frac{5}{16})$
$f (x) = k (x^{2} - \frac{21}{8} x + \frac{5}{16}) = \frac{k}{16} (16 x^{2} - 42 x + 5) = \frac{k}{16} (16 x^{2} - 40 x - 2 x + 5) = \frac{k}{16} (16 x^{2} - 2 x - 40 x + 5) = \frac{k}{3} (2 x (8 x - 1) - 5 (8 x - 1)) = \frac{k}{3} (8 x - 1) (2 x - 5)$
Now, the zeroes are given by f(x) = 0.
Thus, $x = \frac{1}{8} and x = \frac{5}{2}$
(iii) We have, sum of zeroes = $- 2 \sqrt{3}$ and product of zeroes = −9.
So, the required quadratic polynomial will be $f (x) = k (x^{2} + 2 \sqrt{3} x - 9)$ .
$f (x) = k (x^{2} + 2 \sqrt{3} x - 9) = k (x^{2} + 3 \sqrt{3} x - \sqrt{3} x - 9) = k (x + 3 \sqrt{3}) (x - \sqrt{3})$
Now, the zeroes are given by f(x) = 0.
Thus, $x = - 3 \sqrt{3} and x = \sqrt{3}$ .

(iv) We have, sum of zeroes = $\frac{- 3}{2 \sqrt{5}}$ and product of zeroes = $\frac{- 1}{2}$
So, the required quadratic polynomial will be $f (x) = k (x^{2} + \frac{3}{2 \sqrt{5}} x - \frac{1}{2})$ .
$f (x) = \frac{k}{2 \sqrt{5}} (2 \sqrt{5} x^{2} + 3 x - \sqrt{5}) = \frac{k}{2 \sqrt{5}} (2 \sqrt{5} x^{2} + 5 x - 2 x - \sqrt{5}) = \frac{k}{2 \sqrt{5}} [\sqrt{5} x (2 x + \sqrt{5}) - 1 (2 x + \sqrt{5} x)] = \frac{k}{2 \sqrt{5}} (2 x + \sqrt{5}) (\sqrt{5} x - 1)$

Now, the zeroes are given by f(x) = 0.
Thus, $x = \frac{- \sqrt{5}}{2} and x = \frac{1}{\sqrt{5}}$ .

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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 factorization.

(i) $- \frac{8}{3}, \frac{4}{3}$ (ii) $\frac{21}{8}, \frac{5}{16}$ (iii) $- 2 \sqrt{3}, - 9$ (iv) $\frac{- 3}{2 \sqrt{5}}, - \frac{1}{2}$

Q. Question 1 (i)
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 factorization.
i)

\frac{- 8}{3}, \frac{4}{3}

Q. Question 1 (ii)
For each of the following, find a quadratic polynomial whose sum and product of the zeroes respectively are as given. Also, find the zeroes of these polynomials by factorization.

iii)

- 2 \sqrt{3}, - 9

Q. Question 1 (iv)
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 factorization.
iv)

= \frac{- 3}{2 \sqrt{5}}, - \frac{1}{2}

Q. Question 1 (ii)
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 factorization.
ii)

\frac{21}{8}, \frac{5}{16}

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