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

Factoring Cubic Polynomials

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

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