What are the possible rational roots of #2x^3+3x^2-8x+3=0# and then determine the rational roots?
1 Answer
The "possible" rational roots are:
#+-1/2, +-1, +-3/2, +-3#
The actual roots are:
#1, 1/2, -3#
Explanation:
Given:
#2x^3+3x^2-8x+3=0#
By the rational roots theorem, any rational zeros of this cubic are expressible in the form
That means that the only possible rational zeros are:
#+-1/2, +-1, +-3/2, +-3#
In addition, note that the sum of the coefficients is
#2+3-8+3 = 0#
Hence
#0 = 2x^3+3x^2-8x+3#
#color(white)(0) = (x-1)(2x^2+5x-3)#
To find the zeros of the remaining quadratic we could try each of the other possible rational roots in turn, but I'd rather complete the square:
#0 = 8(2x^2+5x-3)#
#color(white)(0) = 16x^2+40x-24#
#color(white)(0) = (4x)^2+2(4x)(5)+(5)^2-49#
#color(white)(0) = (4x+5)^2-7^2#
#color(white)(0) = ((4x+5)-7)((4x+5)+7)#
#color(white)(0) = (4x-2)(4x+12)#
#color(white)(0) = (2(2x-1))(4(x+3))#
#color(white)(0) = 8(2x-1)(x+3)#
So the remaining two roots are:
#x=1/2" "# and#" "x=-3#