90 lines
1.8 KiB
Typst
90 lines
1.8 KiB
Typst
#import "../handout.typ": *
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#import "../macros.typ": *
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#import "@preview/cetz:0.3.1"
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= Tropical Cubic Polynomials
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#problem()
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Consider the polynomial $f(x) = x^3 #tp x^2 #tp 3x #tp 6$. \
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- sketch a graph of this polynomial
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- use this graph to find the roots of $f$
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- write (and expand) a product of linear factors with the same graph as $f$.
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#graphgrid(none)
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#v(1fr)
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#pagebreak() // MARK: page
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#problem()
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Consider the polynomial $f(x) = x^3 #tp x^2 #tp 6x #tp 6$. \
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- sketch a graph of this polynomial
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- use this graph to find the roots of $f$
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- write (and expand) a product of linear factors with the same graph as $f$.
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#graphgrid(none)
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#v(1fr)
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#problem()
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Consider the polynomial $f(x) = x^3 #tp 6x^2 #tp 6x #tp 6$. \
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- sketch a graph of this polynomial
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- use this graph to find the roots of $f$
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- write (and expand) a product of linear factors with the same graph as $f$.
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#graphgrid(none)
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#v(1fr)
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#pagebreak() // MARK: page
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#problem()
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If $f(x) = a x^3 #tp b x^2 #tp c x #tp d$, then $accent(f, macron)(x) = a x^3 #tp B x^2 #tp C x #tp d$ for some $B$ and $C$. \
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Using the last three problems, find formulas for $B$ and $C$ in terms of $a$, $b$, $c$, and $d$.
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#v(1fr)
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#pagebreak() // MARK: page
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#problem()
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What are the roots of the following polynomial?
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#align(
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center,
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box(
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inset: 3mm,
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$
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3 x^6 #tp 4 x^5 #tp 2 x^4 #tp x^3 #tp x^2 #tp 4 x #tp 5
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$,
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),
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)
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#v(1fr)
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#pagebreak() // MARK: page
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#problem()
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If
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$
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f(x) = c_0 #tp c_1 x #tp c_2 x^2 #tp ... #tp c_n x^n
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$
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then
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$
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accent(f, macron)(x) = c_0 #tp C_1 x #tp C_2 x^2 #tp ... #tp C_(n-1) x^(n-1) #tp c_n x^n
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$
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#v(2mm)
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Find a formula for each $C_i$ in terms of $c_0, c_1, ..., c_n$. \
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Then, find formulas for the roots $r_1, r_2, ..., r_n$.
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#v(1fr)
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#problem()
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Can you find a geometric interpretation of these formulas \
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in terms of the points $(-i, c_i)$ for $0 <= i <= n$?
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#v(0.5fr)
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