A POLYNOMIAL is a monomial or a sum of monomials. 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

What is the degree and leading coefficient of 3x5 – 3x + 2 ? 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

EVALUATING A POLYNOMIAL FUNCTION 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

EVALUATING A POLYNOMIAL FUNCTION 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

EVALUATING A POLYNOMIAL FUNCTION Find f(m + 2) if f(x) = 3x2 – 2x – 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

EVALUATING A POLYNOMIAL FUNCTION Find 2g(-2a) if g(x) = 3x2 – 2x – 6 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 f(x) = 3 Constant Function Degree = 0 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0 Max. Zeros: 0

POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 Max. Zeros: 1

POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 Max. Zeros: 2

POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Max. Zeros: 3

POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function Degree = 4 Max. Zeros: 4

POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic Function Degree = 5 Max. Zeros: 5

POLYNOMIAL FUNCTIONS f(x) = x2 Degree: Even Leading Coefficient: + END BEHAVIOR f(x) = x2 Degree: Even Leading Coefficient: + End Behavior: As x  -∞; f(x)  +∞ As x  +∞; f(x)  +∞

POLYNOMIAL FUNCTIONS f(x) = -x2 Degree: Even Leading Coefficient: – END BEHAVIOR f(x) = -x2 Degree: Even Leading Coefficient: – End Behavior: As x  -∞; f(x)  -∞ As x  +∞; f(x)  -∞

POLYNOMIAL FUNCTIONS f(x) = x3 Degree: Odd Leading Coefficient: + END BEHAVIOR f(x) = x3 Degree: Odd Leading Coefficient: + End Behavior: As x  -∞; f(x)  -∞ As x  +∞; f(x)  +∞

POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: – END BEHAVIOR f(x) = -x3 Degree: Odd Leading Coefficient: – End Behavior: As x  -∞; f(x)  +∞ As x  +∞; f(x)  -∞

Complex Numbers Note that squaring both sides yields: therefore and so And so on…

Real numbers and imaginary numbers are subsets of the set of 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

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. Sum: Difference:

Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i 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

The Fundamental Theorem of Algebra 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 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 Linear Factorization Theorem 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. The Linear Factorization Theorem If f (x) = anxn + an-1xn-1 + … + 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.

Find all the zeros of Solutions: The possible rational zeros are Synthetic division or the graph can help: 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)]

Factoring Cubic Polynomials

Find the Greatest Common Factor Identify each term in the polynomial. 14x3 – 21x2 2•7•x•x•x 3•7•x•x – Identify the common factors in each term GCF = 7x2 The GCF is? Use the distributive property to factor out the GCF from each term 14x3 – 21x2 = 7x2(2x – 3)

Factor Completely 4x3 + 20x2 + 24x 2•2•x•x•x 2•2•5•x•x 2•2•2•3•x Identify each term in the polynomial. 4x3 + 20x2 + 24x Identify the common factors in each term 2•2•x•x•x 2•2•5•x•x 2•2•2•3•x + + GCF = 4x The GCF is? 4x3 + 20x2 + 24x = 4x(x2 + 5x +6) Use the distributive property to factor out the GCF from each term 4x (x + 2)(x + 3)

Factor by Grouping x3 - 2x2 - 9x + 18 = (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 = (x – 2)(x2 – 9) Use the distributive property = (x – 2)(x – 3)(x + 3) Factor the difference of two squares

Sum of Two Cubes Pattern a3 + b3 = (a + b)(a2 - ab + b2) Example Now, use the pattern to factor x3 + 27 = x3 + 3•3•3 = x3 + 33 x3 + 33 = (x + 3)(x2 - 3x + 32) = (x + 3)(x2 - 3x + 9) So x3 + 27 = (x + 3)(x2 - 3x + 9)

Difference of Two Cubes Pattern a3 - b3 = (a - b)(a2 + ab + b2) Example Now, use the pattern to factor n3 - 64 = n3 - 4•4•4 = n3 - 43 n3 - 43 = (n - 4)(n2 + 4n + 42) = (n - 4)(n2 + 4n + 16) So n3 - 64 = (n - 4)(n2 + 4n + 16)