How do you find the x and y intercepts for f(x) = pi(x)^5 + pi(x)^4 + sqrt{3}(x) + 1?

Oct 18, 2015

$y$ intercept is just $\left(0 , f \left(0\right)\right) = \left(0 , 1\right)$
$x$ intercept can be determined numerically as $\approx \left(- 0.71443260387 , 0\right)$

Explanation:

$f \left(x\right) = \pi {x}^{5} + \pi {x}^{4} + \sqrt{3} x + 1$

The $y$ intercept is simply $f \left(0\right) = 1$

The $x$ intercept (of which there is exactly one), is the root of $f \left(x\right) = 0$.

Since all of the coefficients of $f \left(x\right)$ are positive and the leading term is an odd power of $x$, there is exactly one root.

$f \left(- 1\right) = - \pi + \pi - \sqrt{3} + 1 = 1 - \sqrt{3} < 0$
$f \left(0\right) = 1 > 0$

So the root lies somewhere in $\left(- 1 , 0\right)$

It is extremely difficult to find algebraically, but we can approximate using Newton's method.

Let our first approximation be $- 0.5$ and iterate using the formula:

${a}_{i + 1} = {a}_{i} - f \frac{{a}_{i}}{f ' \left({a}_{i}\right)}$

$f ' \left(x\right) = 5 \pi {x}^{4} + 4 \pi {x}^{3} + \sqrt{3}$

Putting these formulae into a spreadsheet, I got:

${a}_{0} = - 0.5$
${a}_{1} = - 0.70310491938$
${a}_{2} = - 0.71152815904$
${a}_{3} = - 0.71143261702$
${a}_{4} = - 0.71443260387$
${a}_{5} = - 0.71443260387$