How do you find the number of possible positive real zeros and negative zeros then determine the rational zeros given #f(x)=x^3+7x^2+7x-15#?
1 Answer
Use Descartes' Rule of Signs to find that
Further find all (rational) zeros:
Explanation:
Given:
#f(x) = x^3+7x^2+7x-15#
Descartes' Rule of Signs
The pattern of signs of the coefficients is
The pattern of signs of coefficients of
Rational Roots Theorem
Since
That means that the only possible rational zeros are:
#+-1, +-3, +-5, +-15#
Sum of coefficients shortcut
The sum of the coefficients of
#1+7+7-15 = 0#
Hence
#x^3+7x^2+7x-15 = (x-1)(x^2+8x+15)#
Note that
#x^2+8x+15 = (x+3)(x+5)#
So the remaining two zeros are