SOLUTION: Hello, I need help with the following problem: Sketch the graph of f(x)=(x-1)^3(x+2)^2(x-3). What is the behavior of f near each zero? What is the end-behavior of f itself?

Question 454986: Hello, I need help with the following problem:
Sketch the graph of f(x)=(x-1)^3(x+2)^2(x-3). What is the behavior of f near each zero? What is the end-behavior of f itself?
Find intercepts and set up a sign graph.
Write a polynomial equation of the last possible degree, and leading coefficient one, which has real coefficients, and {2,-2,i} as its zeros.
Thank you so much.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Sketch the graph of f(x)=(x-1)^3(x+2)^2(x-3).

------------------------------------------------------------
What is the behavior of f near each zero?
f(0) = (-1)^3*(2^2)*(-3) = 12
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What is the end-behavior of f itself?
The highest value of x is x^6
As x goes to -oo, y goes to +oo
As x goes to +oo, y goes to +oo
----------------
Find intercepts and set up a sign graph.
Y-int: 12
x-int: x = 1, x = -2, x = 3
==================================
Write a polynomial equation of the least possible degree, and leading coefficient one, which has real coefficients, and {2,-2,i} as its zeros.
--
If I is a root, so is -i.
f(x) = (x-2)(x+2)(x+i)(x-i)
f(x) = (x^2-4)(x^2+1)
f(x) = x^4-5x^2-4
======================
Cheers,
Stan H.
============

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