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Mathematics Dictionary
Dr. K. G. Shih
Polynomial and Function
Subjects


  • AL 02 00 | - Outlines
  • AL 02 01 | - Defintion of polynomial
  • AL 02 02 | - Names of polynomial
  • AL 02 03 | - Two equal polynomials
  • AL 02 04 | - Addition of two polynomials
  • AL 02 05 | - Multiplication of two polynomials
  • AL 02 06 | - Division of two polynomials
  • AL 02 07 | - Defintion of Functions
  • AL 02 08 | - Name of functions
  • AL 02 09 | - Properties of functions
  • AL 02 10 | - Composite of functions
  • AL 02 11 | - Factors of Expressions
  • AL 02 12 | - Inverse functions
  • AL 02 13 | - Rationalize denominator
  • AL 02 14 | - Rational function
  • AL 02 15 | - Ir-rational function
  • AL 02 16 | - Domain and range
  • AL 02 17 | - References
    • Answers


    AL 02 01. Defintions

    Polynomials
    • An algebraic expression an*x^n + ..... + a3*x^3 + a2*x^2 + a1*x + a0
    • Degree is n where n is integer.
    • Coefficients are an, ..... a3,a2, a1, a0.
    Symbol defintion on computer
    • a0 is a subscript 0, a1 is a subscript 1, etc.
    • * is multiplication sign. E.G. 2*3 = 6.
    • ^ is power sign. E.G. 3^2 = 9, 4^3 = 64, etc

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    AL 02 02. Name of polynomials

    By degrees
    • 1. Constant polynomial ............ degree n = 0.
    • 2. Linear polynomial .............. degree n = 1.
    • 3. Quadratic polynomial ........... degree n = 2.
    • 4. Cubic .......................... degree n = 3.
    • 5. Quartic polynomial ............. degree n = 4.
    By coefficients
    • Integral polynomial : Coefficients are integers
    • Rational polynomial : Coefficients are non-integers
    • Ir-ration polunomial :
    • Complex polynomial :

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    AL 02 03. Two polynomials are equal

    Conditions
    • 1. The degree must be the same.
    • 2. The coefficient of each term of two polynomials must be equal.

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    AL 02 04. Addition of two polynomials

    Rules
    • 1. If there is a missing term, we should use 0 coefficient.
    • 2. The power of each term of the polynomials must be line up
    Example : (x + 1) + (2*x^2 + 1)
    • 2*x^2 + 0*x + 1
    • 0*x^2 + 1*x + 1
    • ---------------
    • 2*x^2 + 1*x + 2

    Go to Begin

    AL 02 05. Multiplication of two polynomials

    Rules
    • 1. If there is a missing term, we should use 0 coefficient.
    • 2. The power of each term of the polynomials must be line up
    Example : Prove that (x + 1)*(x^2 - x + 1) = x^3 + 1
    • ................. 1 -1 +1 Coeff of (x^2 + x + 1)
    • ................... +1 +1 Coeff of (x + 1)
    • ............... ---------
    • ................. 1 -1 +1
    • .............. 1 -1 +1
    • .............. ----------
    • .............. 1 +0 +0 +1 (coeff of product)
    • Hence product = x^3 + 0*x^2 + 0*x + 1 = x^3 + 1

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    AL 02 06. Division of two polynomials

    Rules
    • 1. If there is a missing term, we should use 0 coefficient.
    • 2. The power of each term of the polynomials must be line up
    F(x) = (x^7 + 2*x^6 -5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1)/(x^2 + x + 1)
    • Step 1 : Write the coeff of denominator and numerator as below
      • +1 +2 -5 -13 -13 -05 +2 +1 | 1 + 1 + 1
    • Step 2 : Write the first term of quotient as
      • ..... +1 (1st term of quotient)
      • --------------------------
      • +1 +2 -5 -13 -13 -05 +2 +1 | 1 + 1 + 1
      • +1 +1 +1
      • ------------
      • +0 +1 -6 -13 -13 -05 +2 + 1 (Subtract above terms)
    • Step 3 : Write the 1st and 2nd term of quotient as
      • ..... +1 +01 (1st term and 2nd term of quotient)
      • --------------------------
      • +1 +2 -5 -13 -13 -05 +2 +1 | 1 + 1 + 1
      • +1 +1 +1
      • ------------
      • +0 +1 -6 -13 -13 -05 +2 + 1 (Subtract above terms)
      • .. +1 +1 +01
      • -------------
      • ... 0 -7 -14 -13 -05 +2 + 1 (Subtract above terms)
    • Step 4 : All terms of of quotient
      • ..... +1 +01 -07 -07 +1 +1 (Coeff of Quotient)
      • --------------------------
      • +1 +2 -5 -13 -13 -05 +2 +1 | 1 + 1 + 1
      • +1 +1 +1
      • ------------
      • +0 +1 -6 -13 -13 -05 +2 +1 (Subtract above terms)
      • .. +1 +1 +01
      • -------------
      • ... 0 -7 -14 -13 -05 +2 +1 (Subtract above terms)
      • ..... -7 -07 -07
      • -------------------
      • ...... 0 -07 -06 -05 +2 +1 (Subtract above terms)
      • ......... -7 -07 -07
      • -----------------------
      • .......... 0 +01 +02 +2 +1 (Subtract above terms)
      • ............. +1 +1 +1
      • -----------------------
      • .............. 0 1 +1 +1 (Subtract above terms)
      • .................. 1 +1 +1
      • --------------------------
      • .................. 0 +0 + 0 (Subtract above terms and Remainder is 0)
    • Remainder is R(x) = 0*x^2 + 0*x + 0 = 0
    • Hence F(x) = (x^5 + x^4 -7*x^3 - 7*x^2 + x + 1)
    Example : Prove that (x + 1) is a factor of x^5 + x^4 -7*x^3 - 7*x^2 + x + 1.
    • Use synthetic division
    • Put the coefficients 1,1,-7,-7,1,1 and (x + 1) as -1 at first line as below
    • Put 1st coefficient in 3rd line
    • Put 1*(-1) = -1 under 2nd coefficient at 2nd line
    • Add 2nd coefficient +1 with -1 and we get zero (put at 3rd line)
    • Then 0*(-1) = 0 put at 2nd line under 3rd coefficient -7
    • Then put -7 + 0 = - 7 at 3rd line
    • Repeat this procedures
    • The last number at 3rd line is zero (Remainder = 0)
    • Then (x + 1) is a factor
    • Example as below :
    • +1 +1 -7 -7 +1 +1 | -1 .... Coefficients and -1 at first line
    • .. -1 -0 +7 -0 -1 ......... The 2nd line
    • ------------------------
    • +1 +0 -7 +0 +1 +0 ......... 3rd line add above lines
    • The number at 3rd lines are coefficients of quotient
    • The last number at 3rd line is the remainder
    • Since remainder is zero, hence (x+1) is factor
    • Hence F(x) = (x + 1)*(x^4 - 7*x^2 +1)
    Example Is (x - 1) a factor of x^5 + x^4 -7*x^3 - 7*x^2 + x + 1 ?
    • +1 + 1 -7 -7 +1 +1 | +1
    • .. + 1 +2 -5-12-11
    • ---------------------
    • +1 + 2 -5-12-11+10
    • Since the remainder is 10, hence (x + 1) is not a factor

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    AL 02 07. Defintion of Functions
    • An expression has one variable has replation with other variable.
    • Expression : y = F(x).
      • Independent variable is x. It is also called domain.
      • Dependent variable is y. It is also called range.
    • For y = F(x), it has only one y value for each given x value.
      • y = Sqr(1 - x^2) is a function.
      • x^2 + y^2 = 1 is not a functione because for each x value y has 2 values.

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    AL 02 08. Name of functions

    By degree
    • Constant function : y = c where c is constant. It is a horizontal line
    • Linear function : y = a*x + b. It is a line.
    • Quadratic functions : y = a*x^2 + b*x + c. It is a parabola
    • Cubic function : y = a*x^3 + b*x^2 + c*x + d.
    • Quartic function : y = a*x^4 + b*x^3 + c*x^2 + d*x + e.
    By expression form
    • Explicit function :
      • Function y = F(x) is explicit function.
      • Example : y = x^2 - 6*x + 8 is explicit function.
    • Implicit function :
      • F(x,y) = 0 is implicit funtion.
      • Example : x^2 - 6*x + 8 - y = 0 is implicit funtion.
    • Rational function
      • y = F(x)/G(x).
      • Example : (x+2)/(x^2-6*x+8)
    • Irrational function
      • y = Sqr(F(x)).
      • Example : y = Sqr(1 - x^2)
    • Complex function
      • z = x + i*y.
    • Composite function
      • F(x) is a function and G(x) is a function.
      • F(G(x)) is composite function
      • ln(e^x) = x
      • arcsin(sin(A)) = A
    Diagrams of functions
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    AL 02 09. Properties of functions

    Odd functions
    • If F(-x) = -F(x), F(x) is odd function.
    • Example : F(x) = x^3 is odd function since F(-x) = (-x)^3 = -x^3 = -F(x).
    • Property : Rotate 180 degree about origin, the curve will be same.
    Even functions
    • If F(-x) = F(x), F(x) is evend function.
    • Example : F(x) = x^2 is even function since F(-x) = (-x)^2 = x^2 = F(x).
    • Property : The curve is symmetrical to y-axis.
    Periodic functions
    • If F(x+p) = F(x), F(x) is periodic function with period p.
    • Example 1 : sin(x + 2*pi) = sin(x) and period is 2*pi.
    • Example 2 : cos(x + 2*pi) = cos(x) and period is 2*pi.
    • Example 3 : tan(x + 1*pi) = tan(x) and period is 1*pi.

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    AL 02 10. Composite function

    Defintion
    • y = F(G(x)) is composite function where G(x) is also a fuction.
    • Example : F(x) = x^2 + 2*x - 4 and G(x) = x + 1, find F(G(a))
      • F(G(a)) = (G(a))^2 + 2*G(a)x - 4
      • F(G(a)) = (a + 1)^2 + 2*(a+1) - 4
    Composite of inverse
    • e^(ln(x)) = x.
    • ln(e^x) = x.
    • sin(arcsin(x)) = x.
    • arcsin(sin(A)) = A.

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    AL 02 11. Factors of Expressions

    Formula
    • x^2 - y^2 = (x+y)*(x-y) This is square differen equals sum times difference.
    • x^2 + y^2 = no real factors.
    • x^3 - y^3 = (x-y)*(x^2 + x*y +y^2).
    • x^3 + y^3 = (x+y)*(x^2 - x*y +y^2).
    • x^4 - y^4 = (x+y)*(x-y)*(x^2+y^2).
    • x^4 + y^4 = no real factors.
    • x^5 - y^5 = (x-y)*(x^4 + (x^3)*y +(x^2)*(y^2) - x*(y^3) + y^3.
    • x^5 + y^5 = (x+y)*(x^4 - (x^3)*y +(x^2)*(y^2) + x*(y^3) + y^3..
    Factors in perfect square
    • x^2 - 2*x*y + y^2 = (x-y)^2
    • x^2 + 2*x*y + y^2 = (x+y)^2
    Factors in perfect cube
    • x^3 - 3*(x^2)*y + 3*x*(y^2) - y^3 = (x-y)^3
    • x^3 + 3*(x^2)*y + 3*x*(y^2) + y^3 = (x+y)^3
    Studys Subject | Factors of expression of linear functions

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    AL 02 12. Inverse functions

    Inverse of linear functions     Inverse of Quadratic functions
    • Studys Subject | Inverse of Quadratic functions
    • Defintion
      • y = a*x^2 + b*x + c
      • Inverse : x = a*y^2 + b*y + c

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    AL 02 13. Rationalize denominator

    Example 1 : Rationalize y = 1/Sqr(1-x^2)
    • Make denominator a rational function.
    • Hence multiply numerator and denominator by Sqr(1-x^2).
    • y = Sqr(1-x^2)/(Sqr(1-x^2)*Sqr(1-x^2).
    • y = Sqr(1-x^2)/(1-x^2).
    Example 1 : Rationalize y = 1/(2+i)
    • Make denominator a rational function.
    • Hence multiply numerator and denominator by (2-i).
    • y = (2+i)/((2+i)*(2-i)).
    • y = (2+i)/(2^2 - i^2).
    • Since i^2 = -1.
    • Hence y = (2+i)/5.

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    AL 02 14. Rational functions

    Diagrams
    • Studys Subject | Sketch y = F(x)/G(x)
      • Click start.
      • Click rational function in upper box
      • Programs in lower box
        • Program 1 : y = 1/(a*x^2 + b*x + c).
      • Click program in lower box
    • Studys Subject | Complex numbers

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    AL 02 15. Ir-rational function

    Definition
    • Function including square root
    • Example y = Sqr(x^2 - 1)

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    AL 02 16. Domain and Range
    Definition
    • Domain does not exist if y = infinite. E.G. y = 1/(x-1) and no domain at x = 1
    • For y = Sqr(x), no domain if x less than zero in real number system
    • For y = ln(x), no domain if x less or equal zero in real number system

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    AL 02 17. Reference

    Sketch on computer
    • Studys Subject | Sketch y = F(x) or y = F(x)/G(x)
      • Click start.
      • Click Subject in upper box
        • Subject 1 : Linear functions.
        • Subject 2 : Quadratic functions.
        • Subject 3 : Rational functions.
      • Click program in lower box
    Complex number Demo sketch
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