Transcript tictactoe_phaseII_powerpoint_group1-7 - 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
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 = 1x² + 24
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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