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

11

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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16

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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19

20

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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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