Download 2.2 Power Point Notes

1.
Solve: 2x3 – 4x2 – 6x = 0. (Check with GUT)
2.
Solve algebraically or graphically:
– 2x – 15> 0
x2
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We Know:
f(x) = c
constant
f(x) = mx + b
f(x) =
2
ax
linear
+ bx + c
quadratic
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Polynomial Functions
Pre-Cal






Determine end behavior
Factor a polynomial function
Graph a polynomial function
Find the zeros of a polynomial function
Write a polynomial function given its
zeros
Use GUT to graph and solve polynomial
function
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f(x) = an
n
x
+ an-1
n-1
x
+ ... + a1
1
x
+ a0
where an ≠ 0
Example:
f(x) = 3x4 – 2x3 + 5x – 4
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Standard Form means that the polynomial
is written in _____________Descending
order of
Exponents
_____________
A function of degree “n” has at most “n”
zeros.
If the degree of a function is “n”, then the
number of total zeros (real or nonreal) is
n. (FTA)
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F(x) = a(x – b)(x – c)(x – d)…
Once a polynomial is factored is easy
to find the zeros.
Factor: (x – b)
Solution/zero: x = b
X-Intercept: (b, 0)
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
Whole numbers therefore
exponents are all ______________
Positive
all __________________

Real numbers
all coefficients are___________________

Leading coefficient
an is called the _____________________

Constant term
a0 is called the _____________________

degree
n is equal to the ____________________
highest
(always the _______________
exponent)
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Standard Form
Example
Degree Name
f(x) = a0
f(x) = a1x1 + a0
f(x) = a2x2 + a1x1 + a0
f(x) = a3x3 + a2x2 + a1x1 + a0
f(x) = a4x4 + a3x3 + a2x2 + a1x1 + a0
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End behavior is what the y values are doing as
the x values approach positive and negative
infinity.
It is written: f(x)
_____ as x
-∞, and
f(x)
_____ as x
∞
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
even
If the degree is __________
the ends of the
same
graph go in the _________
direction.

odd
If the degree is __________
the ends of the
opposite directions.
graph go in the _________

Leading coefficient to see what
Look at the ________________
direction the graph is going in.
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 Even
 Odd
exponent
exponent
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1. f(x) = 3x4 – 2x2 + 5x – 8
D:
4, even
2. f(x) = -x2 + 1
D: 2, even
LC: 3, positive
LC: -1, negative
End Behavior:
End Behavior:
∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ---> -∞
-∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
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3. f(x) = x7 – 3x3 + 2x
D:
4. f(x) = -2x6 + 3x – 7
7, odd
D: 6, even
LC: 1, positive
LC: -2, negative
End Behavior:
End Behavior:
-∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ----> -∞
-∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
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6. f(x) = 4x3 + 5x7 – 2
5. f(x) = -4x3 + 3x8
D:
8, even
D: 7, odd
LC: 3, positive
LC: 5, positive
End Behavior:
End Behavior:
∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ----> -∞
∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
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Single Root:
passes through
Triple Root:
flattens out then
passes through
Double Root:
touches and turns
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Double Root:
Multiplicity of two
Triple Root:
Multiplicity of three
Y = x3 has a
multiplicity of 3 at
x=0
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18
19
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1. y = -x5
2. g(x) = x4 + 1
3. f(x) = (x + 1)3
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1. f(x) = x3 – x2 – 2x
x(x2 – x – 2)
x(x – 2)(x + 1)
x = 0 x = 2 x = -1
x y
-2 -8
- ½ 5/8
1
3
-2
12
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2. f(x) = -2x4 + 2x2
-2x2(x2 – 1)
-2x2(x – 1)(x +
1)
x = x0 yx = 1 x = -1
-2 -24
- ½ 3/8
½
2
3/8
-24
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3. f(x) = 3x4 – 4x3
x3(3x – 4)
x = 0 x = 4/3
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4. f(x) = -2x3 + 6x2 – 9/2x
0 = -2x3 + 6x2 – 9/2x
2(0 = -2x3 + 6x2 – 9/2x )
0 = -4x3 + 12x2 – 9x
0 = -x(4x2 - 12x + 9)
0 = -x(2x – 3)(2x – 3)
x=0
x = 3/2
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2. 1, -4, 5
1. 4, -4, and 1
x = 4 x = -4
x=1
(x – 4)(x + 4)(x – 1)
(x2 – 16)(x – 1)
f(x) = x3–x2–16x+16
x = 1 x = -4 x = 5
(x – 1)(x + 4)(x – 5)
(x2 + 3x – 4)(x – 5)
f(x)=x3–5x2+3x2–15x–4x+20
f(x) = x3 – 2x2 – 19x + 20
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3. 2, √11, -√11
x = 2 x = √11 x = - √11
(x – 2)(x - √11)(x + √11)
(x – 2)(x2 – 11)
f(x) = x3 – 2x2 – 11x + 22
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4. -3, 4i
x = -3, x = 4i, x = -4i
**imaginary zeros always
come in conjugate pairs!!
(x + 3)(x – 4i)(x + 4i)
*do the imaginary first!
5. 8, -i
x = 8, x = -i, x = i
(x – 8)(x + i)(x – i)
(x – 8)(x2 – i2)
(x + 3)(x2 – 16i2)
(x – 8)(x2 + 1)
*remember i2 is -1!
f(x) = x3 – 8x2 + 1x – 8
(x + 3)(x2 + 16)
f(x) = x3 + 3x2 + 16x + 48
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The zero is the x value that would
give you zero for y.
X = 2.3
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The zero is the x
value that would
give you zero for y.
X = 3.3
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 f(x)
=
3
x
+
2
2x
– 8x – 16
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