How do you use the Rational Root Theorem to find all the roots of #x^3 + 9x^2 + 19x – 4 = 0#?

1 Answer
Aug 23, 2016

Answer:

The roots are: #-4# and #(-5+-sqrt(29))/2#

Explanation:

#f(x) = x^3+9x^2+19x-4#

By the rational roots theorem any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a factor of the constant term #-4# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1, +-2, +-4#

Trying each in turn, we find:

#f(-4) = -64+9(16)+19(-4)-4 = -64+144-76-4 = 0#

So #x=-4# is a zero and #(x+4)# a factor:

#x^3+9x^2+19x-4 = (x+4)(x^2+5x-1)#

We can solve the remaining quadratic using the quadratic formula:

#x = (-5+-sqrt(5^2-4(1)(-1)))/(2*1)#

#=(-5+-sqrt(29))/2#