Question #dc14d
1 Answer
Nov 20, 2017
Explanation:
#"given the zeros of a polynomial are "x=a" and "x=b#
#"then "(x-a)" and "(x-b)" are factors of the polynomial"#
#"here "x=sqrt2" and "x=-sqrt2#
#rArr(x-sqrt2)" and "(x+sqrt2)" are factors"#
#"calculate the product of the factors"#
#rArr(x-sqrt2)(x+sqrt2)larrcolor(blue)"expand using FOIL"#
#=x^2cancel(+sqrt2x)cancel(-sqrt2x)-2#
#rArrx^2-2" is a factor of the polynomial"#
#"divide the polynomial by "(x^2-2)#
#color(red)(2x)(x^2-2)cancel(color(magenta)(+4x))cancel(-4x)-x^2+2#
#=color(red)(2x)(x^2-2)color(red)(-1)(x^2-2)cancel(color(magenta)(-2))cancel(+2)#
#"quotient "=color(red)(2x-1)," remainder "=0#
#rArr(2x^3-4x-x^2+2)/(x^2-2)=2x-1#
#"equate "2x-1" to zero"#
#2x-1=0rArrx=1/2#
#"the 3 zeros of the polynomial are "x=1/2,x=+-sqrt2#