Polynomial functions

Polynomial functions

Basic Concepts: What is a polynomial?, Identifying functions, Introduction to quadratic functions
Related Concepts: Basic radical functions, Square root of a function, What is a rational function?

Lessons

A polynomial function is a function in the form:

f(x)=anxn+an−1xn−1+an−2xn−2+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}, an−1{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
  • Introduction
    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+7x4−2x3−12x2+x−10f(x) = 9x^{5}+7x^{4}-2x^{3}-12x^{2}+x-10
    2) p(x)=−23x18+37x15−11x58+6p(x) = -23x^{18}+37x^{15}-11x^{58}+6
  • 1.
    Recognizing a Polynomial Function

    Which of the following are not polynomial functions? Explain.

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

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

    c)
    f(x)=(x+3)(x−3)f(x) = (\sqrt x + 3)(\sqrt x - 3)

    d)
    f(x)=x5+πx−7x2+311f(x) = x^{5}+\pi x-\sqrt7 x^{2}+\frac{3}{11}

  • 2.
    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
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3.
Polynomial Functions
3.1
What is a polynomial function?
3.2
Polynomial long division
3.3
Polynomial synthetic division
3.4
Remainder theorem
3.5
Factor theorem
3.6
Rational zero theorem
3.7
Characteristics of polynomial graphs
3.8
Multiplicities of polynomials
3.9
Imaginary zeros of polynomials
3.10
Determining the equation of a polynomial function
3.11
Applications of polynomial functions
3.12
Solving polynomial inequalities
3.13
Fundamental theorem of algebra
3.14
Descartes' rule of signs
Grade 12 Math