X2-2x-8 find the relationship between the …

Let p(x) = x² - 2x - 8
By the method of splitting the middle term,
{tex}x ^ { 2 } - 2 x - 8 = x ^ { 2 } - 4 x + 2 x - 8{/tex}
{tex}= x ( x - 4 ) + 2 ( x - 4 ) = ( x - 4 ) ( x + 2 ){/tex}
For zeroes of p(x),
p(x) = 0

{tex}\Rightarrow ( x - 4 ) ( x + 2 ) = 0{/tex}
{tex}\Rightarrow x - 4 = 0 \text { or } x + 2 = 0{/tex}
{tex}\Rightarrow x = 4 \text { or } x = - 2{/tex}
{tex}\Rightarrow x = 4 , - 2{/tex}

So, the zeroes of p(x) are 4 and -2.
We observe that, Sum of its zeroes
= 4 + (-2) = 2
{tex}= \frac { - ( - 2 ) } { 1 } = \frac { - \text { (Coefficient of } x ) } { \text { Coefficient of } x ^ { 2 } }{/tex}
Product of its zeroes
{tex}= 4 x ( - 2 ) = - 8 = \frac { - 8 } { 1 } = \frac { \text { Constant term } } { \text { Coefficient of } x ^ { 2 } }{/tex}

Hence, relation between zeroes and coefficients is verified.