4.1cPolynomial Functions

4.1cPolynomial Functions - Polynomial Functions Polynomial...

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Unformatted text preview: Polynomial Functions Polynomial Section 4.1 Defn. of a Polynomial Function Defn. A polynomial function is a function whose rule polynomial is given by a polynomial is f ( x) = an x n + an −1 x n −1 + L + a1 x + a 0 an , an −1 ,K , a1 , a0 where where with an ≠ 0 with are real numbers and n is a nonnegative number. A polynomial that consists of only a constant polynomial term is called a constant polynomial. constant The zero polynomial is the constant The zero polynomial 0. polynomial The exponent of the highest power of x that The appears with nonzero coefficient is the degree of the polynomial. degree The nonzero coefficient of the highest power The of the variable is the leading coefficient. leading First-degree polynomials are called linear First-degree functions. functions. Second-degree polynomial functions are Second-degree called quadratic functions. quadratic Third-degree polynomial functions are called Third-degree cubic functions. cubic Fourth-degree polynomial functions are Fourth-degree called quartic functions. quartic The Division Algorithm The If a polynomial f(x) is divided by a nonzero If f(x) polynomial h(x) then there is a quotient h(x) polynomial q(x) and a remainder polynomial q(x and r(x) such that r(x) Dividend f ( x ) = h( x ) • q ( x ) + r ( x ) Divisor Quotient Remainder where r(x)=0 or r(x) has degree less than the degree of the divisor, h(x). h(x). Remainders and Factors Remainders If the remainder is 0, the divisor and the If quotient are factors of the dividend. quotient If a polynomial f(x) is divided by x - c, then If f(x) then the remainder is f(c). f(c). A polynomial function f(c) has a linear factor polynomial x – a iff f(a) = 0. f(a) Zeros, x-intercepts, solutions, and factors and Let f(x) be a polynomial. If r is a real number f(x) that satisfies any of the following statements, then r satisfies all the statements. satisfies r is a zero of the function f r is an x-intercept of the graph of f -intercept x = r is a solution, or root, of the equation f(x) =0 x – r is a factor of f Number of Zeros Number A polynomial of degree n has at most n polynomial distinct real zeros. distinct Joke Time Joke What would you say if What someone took your playing cards? cards? dec-a-gon What kind of insect is good at math? at an account-ant ...
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