Polynomial functions

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

A polynomial function is a function of the form f ( x ) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n  0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, a n is the leading coefficient , a 0 is the constant term , and n is the degree . a n  0 a n a n leading coefficient a 0 a 0 constant term n n degree descending order of exponents from left to right. n n – 1

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 3x 5 – 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

You are already familiar with some types of polynomial functions. Here is a summary of common types of polynomial functions. 4 Quartic f ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 0 Constant f ( x ) = a 0 3 Cubic f ( x ) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 2 Quadratic f ( x ) = a 2 x 2 + a 1 x + a 0 1 Linear f ( x ) = a 1 x + a 0 Degree Type Standard Form

Polynomial Functions The largest exponent within the polynomial determines the degree of the polynomial. Quartic 4 Cubic 3 Quadratic 2 Linear 1 Name of Function Degree Polynomial Function in General Form

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. S OLUTION The function is a polynomial function. It has degree 4, so it is a quartic function. The leading coefficient is – 3. Identifying Polynomial Functions f ( x ) = x 2 – 3 x 4 – 7 1 2 Its standard form is f ( x ) = – 3 x 4 + x 2 – 7. 1 2

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number. S OLUTION Identifying Polynomial Functions f ( x ) = x 3 + 3 x

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. S OLUTION The function is not a polynomial function because the term 2 x – 1 has an exponent that is not a whole number. Identifying Polynomial Functions f ( x ) = 6 x 2 + 2 x – 1 + x

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. S OLUTION The function is a polynomial function. It has degree 2, so it is a quadratic function. The leading coefficient is  . Identifying Polynomial Functions Its standard form is f ( x ) =  x 2 – 0.5 x – 2. f ( x ) = – 0.5 x +  x 2 – 2

f ( x ) = x 3 + 3 x f ( x ) = 6 x 2 + 2 x – 1 + x Polynomial function? f ( x ) = x 2 – 3 x 4 – 7 1 2 Identifying Polynomial Functions f ( x ) = – 0.5 x +  x 2 – 2

POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(-2) if f(x) = 3x 2 – 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) = 3x 2 – 2x – 6 f(2a) = 3(2a) 2 – 2(2a) – 6 f(2a) = 12a 2 – 4a – 6

POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(m + 2) if f(x) = 3x 2 – 2x – 6 f(m + 2) = 3(m + 2) 2 – 2(m + 2) – 6 f(m + 2) = 3(m 2 + 4m + 4) – 2(m + 2) – 6 f(m + 2) = 3m 2 + 12m + 12 – 2m – 4 – 6 f(m + 2) = 3m 2 + 10m + 2

POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find 2 g (-2a) if g (x) = 3x 2 – 2x – 6 2 g (-2a) = 2[3(-2a) 2 – 2(-2a) – 6 ] 2 g (-2a) = 2[12a 2 + 4a – 6] 2 g (-2a) = 24a 2 + 8a – 12

Examples of Polynomial Functions

Examples of Nonpolynomial Functions

Polynomial functions

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