What are all the possible rational zeros for #f(x)=3x^3-11x^2+5x+3# and how do you find all zeros?
1 Answer
Explanation:
#f(x) = 3x^3-11x^2+5x+3#
By the rational roots theorem, any rational zero of
That means that the only possible rational zeros are:
#+-1/3, +-1, +-3#
Note first that there's a shortcut in this particular example, in that the sum of the coefficients is
Hence we can deduce that
#3x^3-11x^2+5x+3 = (x-1)(3x^2-8x-3)#
We could just try the remaining rational possibilities that we listed, but we can factor the remaining quadratic (
Find a pair of factors of
The pair
Use this pair to split the middle term and factor by grouping:
#3x^2-8x-3 = 3x^2-9x+x-3#
#color(white)(3x^2-8x-3) = (3x^2-9x)+(x-3)#
#color(white)(3x^2-8x-3) = 3x(x-3)+1(x-3)#
#color(white)(3x^2-8x-3) = (3x+1)(x-3)#
Hence the other two zeros of