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

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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#