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
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 ?
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
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
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
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
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
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = 3
Constant Function
Degree = 0
Max. Zeros: 0
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x + 2
LinearFunction
Degree = 1
Max. Zeros: 1
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QuadraticFunction
Degree = 2
Max. Zeros: 2
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CubicFunction
Degree = 3
Max. Zeros: 3
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
QuarticFunction
Degree = 4
Max. Zeros: 4
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
QuinticFunction
Degree = 5
Max. Zeros: 5
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) +∞
f(x) = x2
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: –
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) -∞
f(x) = -x2
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior:
As x -∞; f(x) -∞
As x +∞; f(x) +∞
f(x) = x3
POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: –
End Behavior:
As x -∞; f(x) +∞
As x +∞; f(x) -∞
f(x) = -x3
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…
Real NumbersImaginary Numbers
Real numbers and imaginary numbers are subsets of the set of complex numbers.
Complex Numbers
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
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:
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
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
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
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.
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.
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
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
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
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)
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