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QUESTION

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

x=2" " or " "x=46

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 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 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