Find all zeroes of polynomial $4 x^{4} - 20 x^{3} + 23 x^{2} + 5 x - 6$ , if two of its zeroes are $2$ and $3$ .

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Solution

Solution:
Let $p (x) = 4 x^{4} - 20 x^{3} + 23 x^{2} + 5 x - 6$

If two of its zeroes are $2$ and $3.$

So, $p (x) = 4 x^{4} - 20 x^{3} + 23 x^{2} + 5 x - 6$

$\Rightarrow p (x) = (x - 2) (x - 3) (4 x^{2} - 1)$

$\Rightarrow p (x) = (x - 2) (x - 3) (2 x - 1) (2 x + 1)$

For zeroes of polynomial, $p (x) = 0$

$(x - 2) = 0$ or $(x - 3) = 0$ or $(2 x - 1)$ or $(2 x + 1) = 0$

$x = 2$ or $x = 3$ or $x = \frac{1}{2}$ or $x = \frac{- 1}{2}$

Hence, all zeroes of the polynomial are $2, 3, \frac{1}{2}$ and $\frac{- 1}{2} .$

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