1. Solve: 2x 3 – 4x 2 – 6x = 0. (Check with GUT) 2. Solve algebraically or graphically: x 2 –...
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Transcript of 1. Solve: 2x 3 – 4x 2 – 6x = 0. (Check with GUT) 2. Solve algebraically or graphically: x 2 –...
1. Solve: 2x3 – 4x2 – 6x = 0. (Check with GUT)
2. Solve algebraically or graphically: x2 – 2x – 15> 0
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We Know: f(x) = c
f(x) = mx + b
f(x) = ax2 + bx + c
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constant
linear
quadratic
Pre-Cal
Polynomial Functions
Determine end behaviorFactor a polynomial functionGraph a polynomial function Fin the zeros of a polynomial
functionWrite a polynomial function given its
zerosUse GUT to graph and solve
polynomial function4
f(x) = anxn + an-1xn-1 + ... + a1x1 + a0
where an ≠ 0
Example: f(x) = 3x4 – 2x3 + 5x – 4
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Standard Form means that the polynomial is written in _____________ order of _____________
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)
Descending Exponents
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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 = bX-Intercept: (b, 0)
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exponents are all ______________ therefore all __________________
all coefficients are___________________
an is called the _____________________
a0 is called the _____________________
n is equal to the ____________________ (always the _______________ exponent)
Whole numbersPositive
Real numbers
Leading coefficient
Constant term
degreehighest
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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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If the degree is __________ the ends of the graph go in the _________ direction.
If the degree is __________ the ends of the graph go in the _________ directions.
Look at the ________________ to see what direction the graph is going in.
odd
same
opposite
Leading coefficient
even
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Even exponent
Odd exponent
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1. f(x) = 3x4 – 2x2 + 5x – 8
D:
LC:
End Behavior:
f(x) --->____ as x --->
f(x) --->____ as x ---->
2. f(x) = -x2 + 1
D:
LC:
End Behavior:
f(x) --->____as x --->
f(x) --->____ as x ---->
-∞
∞∞
-∞
∞
∞ -∞
-∞
3, positive
2, even
-1, negative
4, even
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3. f(x) = x7 – 3x3 + 2x
D:
LC:
End Behavior:
f(x) --->____ as x --->
f(x) --->____ as x ---->
4. f(x) = -2x6 + 3x – 7
D:
LC:
End Behavior:
f(x) --->____as x ---->
f(x) --->____ as x ---->
-∞
∞∞
-∞
∞
-∞ -∞
-∞
1, positive
6, even
-2, negative
7, odd
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5. f(x) = -4x3 + 3x8
D:
LC:
End Behavior:
f(x) --->____ as x --->
f(x) --->____ as x ---->
6. f(x) = 4x3 + 5x7 – 2
D:
LC:
End Behavior:
f(x) --->____as x ---->
f(x) --->____ as x ---->
-∞
∞∞
-∞
∞
∞ -∞
∞
3, positive
7, odd
5, positive
8, even
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Single Root:passes through
Double Root:touches and
turns
Triple Root:flattens out then
passes through
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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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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 -23 12
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2. f(x) = -2x4 + 2x2 -2x2(x2 – 1)
-2x2(x – 1)(x + 1) x = 0 x = 1 x = -1
x y
-2 -24- ½ 3/8
½ 3/8
2 -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
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
2. 1, -4, 5
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!
(x + 3)(x2 – 16i2)
*remember i2 is -1!
(x + 3)(x2 + 16)
f(x) = x3 + 3x2 + 16x + 48
5. 8, -i
x = 8, x = -i, x = i
(x – 8)(x + i)(x – i)
(x – 8)(x2 – i2)
(x – 8)(x2 + 1)
f(x) = x3 – 8x2 + 1x – 8
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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
f(x) = x3 + 2x2 – 8x – 16
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