SOLUTION: What is the end behavior of the graph of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4?

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Question 999619: What is the end behavior of the graph of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
What is the end behavior of the graph of the polynomial function?
recall:
All even-degree polynomials behave, on their ends, like quadratics, and all odd-degree polynomials behave, on their ends, like cubics.
consider are the sign and the degree of the leading term;
in your case the exponent says that this is a degree 6 polynomial, so the graph will behave roughly like a quadratic: up on both ends or down on both ends
since the sign on the leading coefficient is positive, the graph will be up on+both ends


+f%28x%29+=+3x%5E6+%2B+30x%5E5+%2B+75x%5E4=>even-degree polynomial with positive leading coefficient which means the graph will behave roughly like a quadratic an will be up on+both ends
now we can find solutions to this function:
+f%28x%29+=+3x%5E4%28x%5E2+%2B+10x+%2B+25%29
+f%28x%29+=+3x%5E4%28x%5E2+%2B+5x%2B5x+%2B+25%29
+f%28x%29+=+3x%5E4%28%28x%5E2+%2B+5x%29%2B%285x+%2B+25%29%29
+f%28x%29+=+3x%5E4%28x%28x+%2B+5%29%2B5%28x+%2B+5%29%29
+f%28x%29+=+3x%5E4%28x+%2B+5%29%5E2
roots:
+0+=+3x%5E4%28x+%2B+5%29%5E2

if +0+=+3x%5E4=>x=4 (multiplicity four)
if +0+=+%28x+%2B+5%29%5E2=>x=-5 (multiplicity two)

+graph%28+600%2C+600%2C+-10%2C+10%2C+-20%2C+1500%2C+3x%5E6+%2B+30x%5E5+%2B+75x%5E4%29+