What are all the possible rational zeros for #f(x)=5x^3-11x^2+7x-1# and how do you find all zeros?
2 Answers
Zeros are
Explanation:
Zeros are
graph{5x^3-11x^2+7x-1 [-10, 10, -5, 5]}
[Ans]
The "possible" rational zeros are:
#+-1/5, +-1#
The actual zeros are:
#1" "# with multiplicity#2#
#1/5" "# with multiplicity#1#
Explanation:
Given:
#f(x) = 5x^3-11x^2+7x-1#
By the rational root theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1/5, +-1#
To find the actual zeros, we could just try each of these in turn, but there is a shortcut:
Note that the sum of the coefficients of
#5-11+7-1 = 0#
Hence
#5x^3-11x^2+7x-1 = (x-1)(5x^2-6x+1)#
The same is true of the remaining quadratic:
#5-6+1 = 0#
Hence
#5x^2-6x+1 = (x-1)(5x-1)#
The remaining linear factor
So the zeros of
#1" "# with multiplicity#2#
#1/5" "# with multiplicity#1#
graph{5x^3-11x^2+7x-1 [-2.107, 2.89, -1.14, 1.36]}