Mathematics: Try It for Example 3: Is ⎯.3333... a rational number? Explain your reasoning. *

Try It for Example 3: Is ⎯.3333... a rational number?

KHaire7013

27.04.2022

option d is correct x = –10, x = –5, or x = 5

step-by-step explanation:

given : function - $f(x)=x^3+10x^2-25x-250$

one root x=-10

the remainder theorem : solve the polynomial through long division.

using formula p(x) = (x – a) q(x) + r(x).

where, p(x) is the dividend , (x-a) is the divisor or root , q(x) is the quotient , r(x) is the remainder

now we solve the given function using remainder theorem

$p(x)=x^3+10x^2-25x-250$ , $q(x)=x+10$

$\frac{p(x)}{x-a}=q(x)+r(x)$

$\frac{ x^3 + 10x^2 - 25x - 250}{x+10}=x^2+(-25x-250)$

$\frac{ x^3 + 10x^2 - 25x - 250}{x+10}=x^2-25+0$

when remainder is zero our quotient = $x^2-25$

i.e, $f(x)=x^3+10x^2-25x-250=(x+10)(x^2-25)$

solve $x^2-25=0$

$x^2=25$

$x=\sqrt25$

$x=\pm5$

therefore, roots of function is -10,5 or-5

$f(x)=x^3+10x^2-25x-250=(x+10)(x-5)(x+5)$

hence, option d is correct x = –10, x = –5, or x = 5

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