What are all the possible rational zeros for #f(x)=x^3+9x^2+15x+7# and how do you find all zeros?
We have two zeros for
We can use the factor theorem here. According to factor theorem if for some
Hence apart from
Hence, we have two zeros for
#f(x) = x^3+9x^2+15x+7#
Any rational zero of
That means that the only possible rational zeros of
It could have other zeros, but they would be irrational or non-Real Complex.
In addition, note that all of the coefficients of
So the only rational zeros we need consider are
#f(-1) = -1+9-15+7 = 0#
#x^3+9x^2+15x+7 = (x+1)(x^2+8x+7)#
#x^2+8x+7 = (x+1)(x+7)#
Notice that the remaining zero is the expected
In this problem, I could find all the roots and then sort out rational
The sum of the coefficients in f(x) is not 0. So,
The sum of the coefficients in f(-x) is 0. So, x + 1 is a factor, and so,
Now, #f(x)=( x + 1 ) (x^2 + ax + b) = x^3 + 9x^2+15x+7).
Comparison of the like coefficients gives a = 8 and b = 7.
The roots of