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

Question

1577 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-10-18T21:28:57

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

$f(x) = pix^5+pix^4+sqrt(3)x+1$

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

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

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

$f(-1) = -pi + pi -sqrt(3) + 1 = 1 - sqrt(3) < 0$
$f(0) = 1 > 0$

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

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:

$a_(i+1) = a_i - f(a_i)/(f'(a_i))$

$f'(x) = 5pix^4+4pix^3+sqrt(3)$

Putting these formulae into a spreadsheet, I got:

$a_0 = -0.5$
$a_1 = -0.70310491938$
$a_2 = -0.71152815904$
$a_3 = -0.71143261702$
$a_4 = -0.71443260387$
$a_5 = -0.71443260387$

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