Question 918970

{{{P(X)=x^3-3x^2-5x+10}}}


A) Evaluate {{{P(x)}}} for integers {{{-3}}} through {{{5}}}:


{{{x=-3}}}

{{{(-3)^3-3(-3)^2-5(-3)+10}}}

{{{-27-27+15+10}}}

{{{-54+25}}}

={{{-29}}}



{{{x=-2}}}

{{{(-2)^3-3(-2)^2-5(-2)+10}}}

{{{-8-12+10+10}}}

{{{-20+20}}}

={{{0}}}


{{{x=-1}}}

{{{(-1)^3-3(-1)^2-5(-1)+10}}}

{{{-1-3+5+10}}}

{{{-4+15}}}

={{{11}}}


{{{x=0}}}

{{{(0)^3-3(0)^2-5(0)+10}}}

{{{0-0+0+10}}}

={{{10}}}



{{{x=1}}}

{{{(1)^3-3(1)^2-5(1)+10}}}

{{{1-3-5+10}}}

{{{-8+11}}}

={{{3}}}



{{{x=2}}}

{{{(2)^3-3(2)^2-5(2)+10}}}

{{{8-12-10+10}}}

{{{-22+18}}}

={{{-4}}}



{{{x=3}}}

{{{(3)^3-3(3)^2-5(3)+10}}}

{{{27-27-15+10}}}

{{{0-15+10}}}

={{{-5}}}



{{{x=4}}}

{{{(4)^3-3(4)^2-5(4)+10}}}

{{{64-48-20+10}}}

{{{74-68}}}

={{{6}}}



{{{x=5}}}

{{{(5)^3-3(5)^2-5(5)+10}}}

{{{125-75-25+10}}}

{{{135-100}}}

={{{35}}}


B) Find all of the zeros of {{{P(x)}}}

{{{P(X)=x^3-3x^2-5x+10}}}

{{{x^3-3x^2-5x+10=0}}}

{{{x^3-5x^2+5x+2x^2-10x+10=0}}}

{{{(x^3+2x^2)-(5x^2+10x)+(5x+10)=0}}}

{{{x^2(x+2)-5x(x+2)+5(x+2)=0}}}

{{{(x+2)(x^2-5x+5) = 0}}}

one zero is:

{{{(x+2) = 0}}}=>{{{highlight(x=-2)}}}

{{{x^2-5x+5 = 0}}} ...use quadratic formula to find other two zeros:

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{x = (-(-5) +- sqrt( (-5)^2-4*1*5 ))/(2*1) }}} 

{{{x = (5 +- sqrt( 25-20 ))/2 }}} 

{{{x = (5 +- sqrt( 5 ))/2 }}} 

{{{x = (5 +- 2.236)/2 }}}

solutions:

{{{x = (5 + 2.236)/2 }}}

{{{x = 7.236/2 }}}

{{{x =3.6}}}

and

{{{x = (5 - 2.236)/2 }}}

{{{x = 2.764/2 }}}

{{{x =1.4}}}

so, zeros are:{{{highlight(x=-2)}}},{{{highlight(x=3.6)}}}, and {{{highlight(x=1.4)}}}



C) Prove that five is an upper bound on the zeros of {{{P(X)}}}


to find bounds for the real roots of the polynomial means we are looking for a number that is greater than all roots (an upper bound) and a number that is less than all roots (a lower bound) 

we are looking if {{{5}}} is greater than all roots
zeros are:
{{{5>highlight(-2)}}},
{{{5>highlight(3.6)}}}, and 
{{{5>highlight(1.4)}}}

so, it's proven that {{{5}}} is an upper bound on the zeros of {{{P(X)}}}


graph of {{{P(X)}}}:


{{{ graph( 600, 600, -20, 20, -20, 20, x^3-3x^2-5x+10) }}}