What are all the rational zeros of #2x^3-15x^2+9x+22#?

1 Answer
Aug 29, 2016

Answer:

Use the rational roots theorem to find the possible rational zeros.

Explanation:

#f(x) = 2x^3-15x^2+9x+22#

By the rational roots theorem, the only possible rational zeros are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #22# and #q# a divisor of the coefficient #2# of the leading term.

So the only possible rational zeros are:

#+-1/2, +-1, +-2, +-11/2, +-11, +-22#

Evaluating #f(x)# for each of these we find that none work, so #f(x)# has no rational zeros.

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We can find out a little more without actually solving the cubic...

The discriminant #Delta# of a cubic polynomial in the form #ax^3+bx^2+cx+d# is given by the formula:

#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#

In our example, #a=2#, #b=-15#, #c=9# and #d=22#, so we find:

#Delta = 18225-5832+297000-52272-106920 = 150201#

Since #Delta > 0# this cubic has #3# Real zeros.

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Using Descartes' rule of signs, we can determine that two of these zeros are positive and one negative.