How do you find the x and y intercepts for #f(x) = x^3 - 2.91x^2 - 7.668x - 3.8151#?

1 Answer
Sep 20, 2015

Answer:

Scale the polynomial to make one with integer coefficients then use the rational root theorem to help find the zeros.

Intercepts are #(0, -3.8151)#, #(-0.9, 0)# (twice) and #(4.71, 0)#

Explanation:

#f(x) = x^3-2.91x^2-7.668x-3.8151#

The intercept with the #y# axis is #(0, f(x)) = (0, -3.8151)#

Let #t = 100/3 x#

Let #g(t) = 1000000/27 f(x) = t^3-97t^2-8520t-141300#

By the rational root theorem, any rational zeros of this polynomial are factors of #141300 = 2^2 3^2 5^2 157#

#g(157) = 3869893 - 2390953 - 1337640 - 141300 = 0#

#g(t)/(t-157) = t^2+60t+900 = (t+30)^2#

So the zeros of #g(t)# are #t = 157# and #t = -30# (twice)

The corresponding values of #x# are #3/100 t#:

#x = 4.71# or #x = -0.9# (twice)

So the intercepts with the #x# axis are #(4.71, 0)# and #(-0.9, 0)# (twice)