Transcript tictactoe_phaseII_powerpoint_group1 - EAmagnet-alg

FACTORING & ANALYZING AND GRAPHING
POLYNOMIALS
Analyzing
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To analyze a graph you must find:
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End behavior
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Max #of turns
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Number of real zeros(roots)
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Critical points
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Graph
End Behavior
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End behavior:
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This describes the far left and right of the graph and
is determined by the coefficient and degree of the
leading term.
The Right side: if the leading coefficient is
positive then the graph goes up to the right but
if it's negative then it falls to the left.
The Left side: if the degree of the leading term
is even then the left remains the same as the
right side but if the term is odd then the left is
opposite.
Example
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y=4x³ – 3x : leading coefficient is 4x³
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4 is positive so the graph goes up to the right and
the degree is 3 which is odd so the left is opposite
of the right and falls on this side.
END BEHAVIOR:
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Right side: f(x)-->+∞ as x--> +∞
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Left side: f(x)--> -∞ as x--> -∞
Max # of Turns and # of Real Roots
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Max Number of Turns: the amount of times the
graph changes direction depends on the degree
of the leading term. It is 1 less than that degree.
Number of Real Roots: if polynomial is an odd
degree it must cross the x-axis at least once. If
it is even then it will cross the x-axis an even
number of times
Derivatives
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Rules:
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the derivative of a constant with 1 term is 0.
If you have a term such as bxª you find the
derivative with this equation (a)(b)xª-1
Critical Points
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Sorry I cant do this part. I kind of need a
refresher on this section so I will wait to get help
instead of trying to do it without understanding
it.
Graph
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Just plug this into your calculator and when
drawing follow these steps:
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Know how many turns you will have
Start from the left and go up to the RELATIVE MAX
then to the RELATIVE MIN—you might have to go
back to the RELATIVE MAX if there are more than
2 turns.
Don't forget that you cross the y-axis on the POINT
OF INFLECTION
Now on to Factoring Polynomials
Synthetic Division
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Easiest way to explain would be to show
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Example: divide 3x³ – 4x² + 5x – 7 by x – 2
2 3 -4 5 -7
6 4 18
3 2 9 11
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Always bring down first coefficient then multiply
this number by the root and write the product
under the next coefficient. Now add while
writing the sums below the addition line.
The numbers below addition line are quotient
and remainder and the degree is always one
less than the degree that you started with:
3x²+2x+9 R of 11
Synthetic Division cont.
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A non example would be:
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Divide x³ – 12 by x – 2
2 1 12
2
1 24
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= 1x² + 24
This is a non example because there were no
place holders added. Two place holders (0)
were needed between the 1 and 12.
Finding Roots/Zeros
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To find the possible rational roots you use p/q
where p=all factors of last term and q= all
factors of leading coefficient.
Example: list all possible rational zeros of each
function 2x³+3x²-8x+3
p=±1, ±3 q=±1, ±2 so the possible zeros
are: ±1, ±1/2, ±3, ±3/2 (used p/q one term
at a time as if you were using FOIL method)
Finding Roots/Zeros cont.
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After you have found the possible rational roots
you must try the possibilities until you find one
with a remainder of 0. (with synthetic division)
Example using the example from last slide: try 1
as a possible root for 2x³+3x²-8x+3
1 2 3 -8 3
2 5 -3
2 5 -3 0 new quotient: 2x²+5x-3
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Use quadratic formula for other 2 roots: -3 & 1/2