Question

verify if -1/2 and 5/2 are zeroes of the polynomial 4x^3-21x-10.if yes then factorize the polynomial​

Answer 1

${ \huge{ \textbf{Given, }}}$

${ \boxed{ \large{ \boxed{p(x) = {4x}^{3} - 21x - 10}}}} \\ { \boxed{ \large{ \boxed{Zeroes \: \: of \: \: p(x) = (- \frac{1}{2}) \: \: \& \: \: \frac{5}{2} }}}}$

Now, Checking whether the given zeroes are applicable to this polynomial or not.

$p( - \frac{1}{2} ) = 4 {( - \frac{1}{2} )}^{3} - 21( - \frac{1}{2} ) - 10 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = 4( - \frac{1}{8} ) + 10.5- 10\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = - \frac{1}{2} + 10.5 - 10 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = - 0.5 - 0.5 = 0$

And,

$p( \frac{5}{2} ) = 4{( \frac{5}{2} )}^{3} - 21( \frac{5}{2} ) - 10\\ \\ \: \: \: \: \: \: \: \: \: \: = 4( \frac{125}{8} ) - \frac{105}{2} - 10\\ \\ \: \: \: \: \: \: \: \: \: \: = \frac{125}{2} - 52.5 - 10\\ \\ \: \: \: \: \: \: \: \: \: \: = 62.5 - 62.5 = 0$

Therefore,

${ \large{ \boxed{ \red{(- \frac{1}{2}) \: \: \& \: \: \frac{5}{2} \: \: are \: \: the \: \: zeroes \: \: of \: \: p(x).}}}}$

Now, Factoring the given polynomial, we get,

$= > {4x}^{3} - 21x - 10 \\ \\ = > {4x}^{3} + {2x}^{2} - {2x}^{2} - x - 20x - 10 \\ \\ = > {4x}^{2} (x + \frac{1}{2} ) - 2x(x + \frac{1}{2} ) - 20(x + \frac{1}{2} )\\ \\ = >(x + \frac{1}{2} )( {4x}^{2} - 2x - 20 ) \\ \\ = > (x + \frac{1}{2} ) [ {4x}^{2} - 10x + 8x - 20 ]\\ \\ = > (x + \frac{1}{2} )[2x(2x - 5) + 4(2x - 5)] \\ \\ = > (x + \frac{1}{2})[(2x - 5)(2x + 4)] \\ \\ = > { \boxed{ \red{(x + \frac{1}{2} )(2x - 5)(2x + 4)}}}$