Polynomial%20Functions

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Polynomial Functions
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A polynomial function is a function of the form
f (x) an x n an 1 x n 1 a 1 x
a 0
Where an ? 0 and the exponents are all whole
numbers.
For this polynomial function, an is the
leading coefficient, a 0 is the constant term,
and n is the degree.
A polynomial function is in standard form if
its terms are written in descending order of
exponents from left to right.
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You are already familiar with some types of
polynomial functions. Here is a summary of common
types ofpolynomial functions.
0
Constant
f (x) a 0
1
Linear
f (x) a1x a 0
2
Quadratic
f (x) a 2 x 2 a 1 x a 0
3
Cubic
f (x) a 3 x 3 a 2 x 2 a 1 x a 0
4
Quartic
f (x) a4 x 4 a 3 x 3 a 2 x 2 a 1 x a 0
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Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is 3.
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Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is not a polynomial function because
the term 3 x does not have a variable base and
an exponentthat is a whole number.
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Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is not a polynomial function because
the term2x 1 has an exponent that is not a
whole number.
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Polynomial function?
f (x) x 3 3x
f (x) 6x2 2 x 1 x
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One way to evaluate polynomial functions is to
usedirect substitution. Another way to evaluate
a polynomialis to use synthetic substitution.
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SOLUTION
2 x 4 0 x 3 (8 x 2) 5 x (7)
Polynomial in standard form
2 0 8 5 7




3
Coefficients
6
18
30
105
35
10
98
2
6
The value of f (3) is the last number you
write, In the bottom right-hand corner.
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GRAPHING POLYNOMIAL FUNCTIONS
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GRAPHING POLYNOMIAL FUNCTIONS
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GRAPHING POLYNOMIAL FUNCTIONS
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Graph f (x) x 3 x 2 4 x 1.
SOLUTION
To graph the function, make a table of values and
plot the corresponding points. Connect the points
with a smooth curve and check the end behavior.
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Graph f (x) x 4 2x 3 2x 2 4x.
SOLUTION
To graph the function, make a table of values and
plot the corresponding points. Connect the points
with a smooth curve and check the end behavior.
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Adding, Subtracting, Multiplying Polynomials
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To or - , or the coeff. of like
terms!Vertical format

• Add 3x32x2-x-7 and x3-10x28.
• 3x3 2x2 x 7 x3
  10x2 8 Line up like terms
• 4x3 8x2 x 1

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Horizontal format Combine like terms

• (8x3 3x2 2x 9) (2x3 6x2 x 1)
• (8x3 2x3)(-3x2 6x2)(-2x x) (9 1)
• 6x3 -9x2 -x
  8
• 6x3 9x2 x 8

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Examples Adding Subtracting

• (9x3 2x 1) (5x2 12x -4)
• 9x3 5x2 2x 12x 1 4
• 9x3 5x2 10x 3
• (2x2 3x) (3x2 x 4)
• 2x2 3x 3x2 x 4
• 2x2 - 3x2 3x x 4
• -x2 2x 4

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Multiplying Polynomials Vertically

• (-x2 2x 4)(x 3)
• -x2 2x 4
  x 3
• 3x2 6x 12 -x3 2x2 4x
  -x3 5x2 2x 12

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Multiplying Polynomials Horizontally

• (x 3)(3x2 2x 4)
• (x 3)(3x2)
• (x 3)(-2x)
• (x
  3)(-4)
• (3x3 9x2) (-2x2 6x) (-4x 12)
• 3x3 9x2 2x2 6x 4x 12
• 3x3 11x2 2x 12
  

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Multiplying 3 Binomials

• (x 1)(x 4)(x 3)
• FOIL the first two
• (x2 x 4x 4)(x 3)
• (x2 3x 4)(x 3)
• Then multiply the trinomial by the binomial
• (x2 3x 4)(x) (x2 3x 4)(3)
• (x3 3x2 4x) (3x2 9x 12)
• x3 6x2 5x - 12

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Some binomial products appear so much we need to
recognize the patterns!

• Sum Difference (SD)
• (a b)(a b) a2 b2
• Example (x 3)(x 3) x2 9
• Square of Binomial
• (a b)2 a2 2ab b2
• (a - b)2 a2 2ab b2

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Last Pattern

• Cube of a Binomial
• (a b)3 a3 3a2b 3ab2 b3
• (a b)3 a3 - 3a2b 3ab2 b3

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Example

• (x 5)3
• a x and b 5
• x3 3(x)2(5) 3(x)(5)2 (5)3
• x3 15x2 75x 125

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Factoring and Solving Polynomial Expressions
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Types of Factoring

• GCF 6x2 15x 3x (2x 5)
• POS x2 10x 25 (x 5)2
• DOS 4x2 9 (2x 3)(2x 3)
• Bustin da B 2x2 5x 12
• (2x2 - 8x) (3x 12)
• 2x(x 4) 3(x 4)
• (x 4)(2x 3)

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Now we will use Sum of Cubes

• a3 b3 (a b)(a2 ab b2)
• x3 8
• (x)3 (2)3
• (x 2)(x2 2x 4)

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Difference of Cubes

• a3 b3 (a b)(a2 ab b2)
• 8x3 1
• (2x)3 13
• (2x 1)((2x)2 2x1 12)
• (2x 1)(4x2 2x 1)

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When there are more than 3 terms use GROUPING

• x3 2x2 9x 18
• (x3 2x2) (-9x 18) Group in twos
• with a
  in the middle
• x2(x 2) - 9(x 2) GCF each
  group
• (x 2)(x2 9)
• (x 2)(x 3)(x 3) Factor all that
  can be
• 
  factored

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Factoring in Quad form

• 81x4 16
• (9x2)2 42
• (9x2 4)(9x2 4) Can anything be
• factored
  still???
• (9x2 4)(3x 2)(3x 2)
• Keep factoring till you cant factor any more!!

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You try this one!

• 4x6 20x4 24x2
• 4x2 (x4 - 5x2 6)
• 4x2 (x2 2)(x2 3)

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Solve

• 2x5 24x 14x3
• 2x5 - 14x3 24x 0 Put
  in standard form
• 2x (x4 7x2 12) 0 GCF
• 2x (x2 3)(x2 4) 0
  Bustin da b
• 2x (x2 3)(x 2)(x 2) 0
  Factor EVERYTHING
• 2x0 x2-30 x20 x-20 set all to zero
  
• x0 xv3 x-2 x2

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Now, you try one!

• 2y5 18y 0
• y0 yv3 yiv3