Polynomial Equation Info
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Polynomial Graph Info
100
What is the end behavior of y = (x + 3)2(x-7)(4x +3)3?
degree 6 with positive leading coefficient ... so up on both ends
100
List the PRZs (possible rational zeros) for the function:
f(x) = 6x^4 - 3x^3 + 5x^2 + 3x - 9
+-1,+-2, +-3,+-9, +-1/2, +-1/3, +-1/6, +-3/2, +-9/2
100
What does the end behavior tell you about the polynomial
The polynomial has an even degree and positive leading coefficient.
200
What are the zeros of y = (x + 3)2(x-7)(4x +3)3?
-3, 7, -3/4
200
How many positive zeros might the function
f(x) = 6x^4 - 3x^3 + 5x^2 + 3x - 9
have?
f(x) term signs: + - + + - ... 3 sign changes so ... 3 or 1 positive zeros
200
What are the obvious zeros of the function?
-3, -1, and 2
300
What is the behavior of each zero in y = (x + 3)2(x-7)(4x +3)3?
-3 = touch because degree 2 (even)
7 = cross because degree 1 (odd)
-3/4 = cross because degree 3 (odd)
300
If
-3+sqrt5i
is a root of a polynomial g(x), then what else has to be a root as well?
-3-sqrt5i
300
What does the behavior of the zeros tell you about the function
-3 = cross so (x + 3) has odd power
-1 = cross so (x + 1) has odd power ... it also crosses kind of like a cubic so maybe (x + 1)3
2 = touch so (x - 2) has even power, maybe (x - 2)2
400
How many turning points could y = (x + 3)2(x-7)(4x +3)3 have?
degree 6 so ... at most 5 turning points ... 5 or less turning points
400
How many negative zeros might the function
f(x) = 6x^4 - 3x^3 + 5x^2 + 3x - 9
have?
f(-x) term signs: + + + - - so one sign change, so exactly 1 negative zero
400
How many turning points does the graph have? What does that tell you about the function equation?
The graph has 3 turning points so the function equation is at least degree 4, maybe higher.
** (-1,0) is NOT a turning point **
500
Sketch a graph of the function y = (x + 3)2(x-7)(4x +3)3 based only on information you know from the factored form of the equation.
end behavior: both ends up
zeros: -3 (touch), 7 (cross), -3/4 (squiggly cross)
500
Test the PRZ x = 3 using synthetic division for the polynomial. What does your result tell you about x = 3?
f(x) = 6x^4 - 3x^3 + 5x^2 + 3x - 9
top row: 6 -3 5 3 9
middle row: -- 18 45 150 459
bottom row: 6 15 50 153 468
This means that x = 3 is not a root of the function f(x)
500
Write the polynomial in factored form based on the information given in the graph.
(x+3)(x+1)3(x-2)2