Polynomial functions

Polynomial functions


A polynomial function is a function in the form:

f(x)=anxn+an1xn1+an2xn2+f\left( x \right)\; = {a_n}{x^n} + \;{a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + +a2x2+a1x+a0 + {a_2}{x^2} + {a_1}x + {a_0}

\bulletcoefficients: an{a_n}, an1{a_{n - 1}}, . . . , a2{a_2}, a1{a_1}
\bulletleading coefficient: "an{a_n}", the coefficient of the highest power of x
\bulletconstant term: "a0{a_0}", the term without xx
\bulletdegree of the polynomial function: nn, the highest power of xx
  • 1.
    Introduction to Polynomial Functions
    \cdot What is a polynomial function?
    \cdot Exercise:
    State the degree, leading coefficient and constant term for the following polynomial functions:
    1) f(x)=9x5+7x42x312x2+x10f(x) = 9x^{5}+7x^{4}-2x^{3}-12x^{2}+x-10
    2) p(x)=23x18+37x1511x58+6p(x) = -23x^{18}+37x^{15}-11x^{58}+6

  • 2.
    Recognizing a Polynomial Function

    Which of the following are not polynomial functions? Explain.

    f(x)=5x2+4x3x1+2f(x) = 5x^{2}+4x-3x^{-1}+2

    f(x)=x3+6x12f(x) = -x^{3}+6x^{\frac{1}{2}}

    f(x)=(x+3)(x3)f(x) = (\sqrt x + 3)(\sqrt x - 3)

    f(x)=x5+πx7x2+311f(x) = x^{5}+\pi x-\sqrt7 x^{2}+\frac{3}{11}

  • 3.
    Classifying Polynomial Functions by Degree

    Complete the chart:

    Complete the chart by classifying polynomial functions by degree

  • 4.
    Classifying Polynomial Functions by Number of Terms
    Write a polynomial satisfying the given conditions:
    i) monomial and cubic
    ii) binomial and linear
    iii) trinomial and quartic