polynomial functions
DESCRIPTION
POLYNOMIAL FUNCTIONS. A POLYNOMIAL is a monomial or a sum of monomials. A POLYNOMIAL IN ONE VARIABLE is a polynomial that contains only one variable. Example: 5x 2 + 3x - 7. POLYNOMIAL FUNCTIONS. The DEGREE of a polynomial in one variable is the greatest exponent of its variable. - PowerPoint PPT PresentationTRANSCRIPT
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POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial or a sum of monomials.
A POLYNOMIAL IN ONE VARIABLE is a polynomial that
contains only one variable.
Example: 5x2 + 3x - 7
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POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient of the term with the highest degree.
What is the degree and leading coefficient of 3x5 – 3x + 2 ?
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POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS
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POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(-2) if f(x) = 3x2 – 2x – 6
f(-2) = 3(-2)2 – 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10
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POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(2a) if f(x) = 3x2 – 2x – 6
f(2a) = 3(2a)2 – 2(2a) – 6
f(2a) = 12a2 – 4a – 6
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POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(m + 2) if f(x) = 3x2 – 2x – 6
f(m + 2) = 3(m + 2)2 – 2(m + 2) – 6
f(m + 2) = 3(m2 + 4m + 4) – 2(m + 2) – 6
f(m + 2) = 3m2 + 12m + 12 – 2m – 4 – 6
f(m + 2) = 3m2 + 10m + 2
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POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find 2g(-2a) if g(x) = 3x2 – 2x – 6
2g(-2a) = 2[3(-2a)2 – 2(-2a) – 6]
2g(-2a) = 2[12a2 + 4a – 6]
2g(-2a) = 24a2 + 8a – 12
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = 3
Constant Function
Degree = 0
Max. Zeros: 0
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x + 2
LinearFunction
Degree = 1
Max. Zeros: 1
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QuadraticFunction
Degree = 2
Max. Zeros: 2
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CubicFunction
Degree = 3
Max. Zeros: 3
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
QuarticFunction
Degree = 4
Max. Zeros: 4
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
QuinticFunction
Degree = 5
Max. Zeros: 5
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) +∞
f(x) = x2
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: –
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) -∞
f(x) = -x2
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) +∞
f(x) = x3
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: –
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) -∞
f(x) = -x3
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1i
Complex Numbers
12 iNote that squaring both sides yields:therefore
and
so
and
iiiii *1* 13 2
1)1(*)1(* 224 iii
iiiii *1*45
1*1* 2246 iiii
And so on…
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Real NumbersImaginary Numbers
Real numbers and imaginary numbers are subsets of the set of complex numbers.
Complex Numbers
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Definition of a Complex Number
If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginary number.
Write the complex number in standard form
81 81 i 241 i 221 i
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Addition and Subtraction of Complex Numbers
If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.
i)db()ca()dic()bia(
i)db()ca()dic()bia(
Sum:
Difference:
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Perform the subtraction and write the answer in standard form.
( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i
234188 i
234298 ii
234238 ii
4
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Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying polynomials and combining like terms.
Perform the operation and write the result in standard form.( 6 – 2i )( 2 – 3i )
F O I L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i
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We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra.
The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n 1, then the equation f (x) 0 has at least one complex root.
The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n 1, then the equation f (x) 0 has at least one complex root.
The Fundamental Theorem of AlgebraThe Fundamental Theorem of Algebra
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The Linear Factorization TheoremThe Linear Factorization Theorem
The Linear Factorization Theorem The Linear Factorization Theorem If f (x) anx
n an1xn1 … a1x a0 b, where n 1 and an 0 , then
f (x) an (x c1) (x c2) … (x cn)
where c1, c2,…, cn are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors.
The Linear Factorization Theorem The Linear Factorization Theorem If f (x) anx
n an1xn1 … a1x a0 b, where n 1 and an 0 , then
f (x) an (x c1) (x c2) … (x cn)
where c1, c2,…, cn are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors.
Just as an nth-degree polynomial equation has n roots, an nth-degree polynomial has n linear factors. This is formally stated as the Linear Factorization Theorem.
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Find all the zeros of 5 4 2( ) 2 8 13 6f x x x x x
Solutions:
The possible rational zeros are 1, 2, 3, 6
Synthetic division or the graph can help:50
40
30
20
10
-10
-20
-30
-40
-50
-4 -3 -2 -1 1 2 3 4
-2 1
Notice the real zeros appear as x-intercepts. x = 1 is repeated zero since it only “touches” the x-axis, but “crosses” at the zero x = -2.
Thus 1, 1, and –2 are real zeros. Find the remaining 2 complex zeros.
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Write a polynomial function f of least degree that has real coefficients, a leading coefficient 1, and 2 and 1 + i as zeros).
Solution:
f(x) = (x – 2)[x – (1 + i)][x – (1 – i)]
3 24 6 4x x x
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Factoring Cubic Polynomials
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Find the Greatest Common Factor
14x3 – 21x2Identify each term in the polynomial.
2•7•x•x•x 3•7•x•x– Identify the common factors in each term
The GCF is?GCF = 7x2
7x2(2x – 3)14x3 – 21x2 =Use the distributive property to factor out the GCF from each term
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Factor Completely
4x3 + 20x2 + 24xIdentify each term in the polynomial.
Identify the common factors in each term2•2•x•x•x 2•2•5•x•x+ 2•2•2•3•x+
GCF = 4x The GCF is?
4x(x2 + 5x +6)4x3 + 20x2 + 24x = Use the distributive property to factor out the GCF from each term(x + 2)(x + 3)4x
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Factor by Grouping
x3 - 2x2 - 9x + 18 Group terms in the polynomial.
= (x3 - 2x2) + (- 9x + 18) Identify a common factor in each group and factor
x•x•x-2•x•x -3•3•x+2•3•3+
= x2(x – 2) + -9(x – 2) Now identify the common factor in each term
Use the distributive property
= (x – 2)(x2 – 9)
Factor the difference of two squares
= (x – 2)(x – 3)(x + 3)
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Sum of Two Cubes Pattern
a3 + b3 = (a + b)(a2 - ab + b2)
x3 + 27 = x3 + 3•3•3 = x3 + 33
Now, use the pattern to factor
x3 + 33 = (x + 3)(x2 - 3x + 32)
= (x + 3)(x2 - 3x + 9)
So x3 + 27 = (x + 3)(x2 - 3x + 9)
Example
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Difference of Two Cubes Pattern
a3 - b3 = (a - b)(a2 + ab + b2)
n3 - 64 = n3 - 4•4•4 = n3 - 43 Now, use the pattern to factor
n3 - 43 = (n - 4)(n2 + 4n + 42)
= (n - 4)(n2 + 4n + 16)
So n3 - 64 = (n - 4)(n2 + 4n + 16)
Example