Answer:
Given polynomial is\[8{{x}^{4}}+8{{x}^{3}}-18{{x}^{2}}-20x-5\]. Since two zeroes are \[\sqrt{\frac{5}{2}}\] and \[-\sqrt{\frac{5}{2}}\] \[\therefore \left( x-\sqrt{\frac{5}{2}} \right)\left( x+\sqrt{\frac{5}{2}} \right)={{(x)}^{2}}-{{\left( \sqrt{\frac{5}{2}} \right)}^{2}}\] \[={{x}^{2}}-\frac{5}{2}\] Dividing the polynomial with \[={{x}^{2}}-\frac{5}{2}\] \[\therefore \,\,8{{x}^{4}}+8{{x}^{3}}-18{{x}^{2}}-20x-5\] \[=\left( {{x}^{2}}-\frac{5}{2} \right)(8{{x}^{2}}+8x+2)\] \[=\left( {{x}^{2}}-\frac{5}{2} \right).2(4{{x}^{2}}+4x+1)\] \[=2\left( {{x}^{2}}-\frac{5}{2} \right)(4{{x}^{2}}+2x+2x+1)\] \[=2\left( {{x}^{2}}-\frac{5}{2} \right)[2x(2x+1)+1(2x+1)]\] \[=2\left( {{x}^{2}}-\frac{5}{2} \right)(2x+1)(2x+1)\] All the zeroes are \[\sqrt{\frac{5}{2}},-\sqrt{\frac{5}{2}},\frac{-1}{2}\] and \[\frac{-1}{2}\].
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