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

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Answer 1 · 2015-09-20T20:09:07

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

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

$f (x) = x^{3} - 2.91 x^{2} - 7.668 x - 3.8151$ $f(x) = x^3-2.91x^2-7.668x-3.8151$

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

Let $t = \frac{100}{3} x$ $t = 100/3 x$

Let $g (t) = \frac{1000000}{27} f (x) = t^{3} - 97 t^{2} - 8520 t - 141300$ $g(t) = 1000000/27 f(x) = t^3-97t^2-8520t-141300$

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

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

$\frac{g (t)}{t - 157} = t^{2} + 60 t + 900 = {(t + 30)}^{2}$ $g(t)/(t-157) = t^2+60t+900 = (t+30)^2$

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

The corresponding values of $x$ $x$ are $\frac{3}{100} t$ $3/100 t$ :

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

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

How do you find the x and y intercepts for f(x) = x^3 - 2.91x^2 - 7.668x - 3.8151f(x)=x3−2.91x2−7.668x−3.8151f(x) = x^3 - 2.91x^2 - 7.668x - 3.8151?