How do you use the rational root theorem to find the roots of #P(x) = 0.25x^2 - 12x + 23#?

1 Answer
Jan 8, 2017

Answer:

#x=2" "# or #" "x=46#

Explanation:

First multiply by #4# to make all of the coefficients into integers:

#4P(x) = 4(0.25x^2-12x+23) = x^2-48x+92#

By the rational root theorem, any rational zeros of #x^2-48x+92# (and therefore of #P(x)#) are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #92# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1, +-2, +-4, +-23, +-46, +-92#

Trying each in turn we soon find:

#P(color(blue)(2)) = 0.25(color(blue)(2))^2-12(color(blue)(2))+23 = 1-24+23 = 0#

So #x=2# is a zero and #(x-2)# a factor:

#x^2-48x+92 = (x-2)(x-46)#

So the other zero of #P(x)# is #x=46#