polynomial

polynomial

1. forms(2)

   • f(x)= d(x)q(x)+r(x) , d(x)=(x-k)
     • Remainder theorem: f(k)=remainder
     • factor theorem: (x-k) is a factor of the polynomial if and only if f(k)=0 then
   • f(x)/d(x)= q(x)+r(x)/d(x), d(x)=(x-k)
     • if f(x)/d(x) is an improper rational expression, then the degree of f(x) is greater than or equal to the degree of d(x)
     • if r(x)/d(x) is a proper rational expression, then the degree of f(x) is less than the degree of d(x)

2. how to find ALL the zeros of the polynomial (find all the REAL zeros first, the rest are COMPLEX zeros)

   • how to find all the REAL zeros of the polynomial
     • easy: factor
     • difficult
       • (condition: interger coefficient) rational zero test, Purpose to find the possible rational zeros of f ,(p/q), p= the factors of the constant, q= the factors of the leading coefficient
       • synthetic division: if the remainder is zero then the divisor k is a real zero of the function, and (x-k) is a factor( factor theorem)
       • (condition: real coeff, a0 != 0), Descartes' rule of signs, Purpose: to find the # of possible positive or negative real zeros of f(x) How to do it: the # of possible positive real zeros of f are either equal to the # of variations in the sign of f(x) or less than that # by an even integer, the # of possible negative real zeros of f are either equal to the # of variations in the sign of f(-x) or less than that # by an even integer
       • (condition: real coefficients and a positive leading coefficient) , purpose: to find the upper and lower bound of all real zeros, how: upper bound:c>0 and the last row of the synthetic divisionis all positive numbers, lower bound : c<0 the numbers in last row of the synthetic division are alternately positive and negative
     • if it is a quadratic, stop and factor or use the quadratic formula
   • complex number n= a+bi (Standard form, a &b are real numbers , i is imaginary unit =sqrt (-1))
     • n= a, real number if b = 0
       • rational(can be expressed as a ratio of two integers.decimal can end) ex. 4= 16/4, 3/5
       • lrrational,,(cannot be expressed as a ratio of two integers.decimal keeps going on and on, ex. pi, sqrt(2), sqrt(3)
     • n=a+bi, imaginary number if a and b != 0
       • n= bi; pure imaginary number , if a =0 and b != 0
   • Complex zeros occur in conjugate pairs, CONDITION: polynomial function f(x) has real coefficients(for example if f(x)=x-i then f(x) does not have real coefficient because i is an imaginary unit), if a +bi where b !=0 , is a zero of the function, then the conjugate a-bi is also a zero of the function
   • Fundamental theorem of Algebra: if f(x)is a polynomial of degree n, n>0, then f has at least one zero in the complex system ex1.(real) f(x)=x-5 x=5, ex2. (imaginary) f(x)=x-i, x=i
   • Linear factorization theorem: if f(x) is a polynomial of degree n, n>0, then f has precisely n linear factors; ex. f(x)=x^3+x=x(x^2+1)=x(x-i)(x+i)
   • factoring a polynomial: every polynomial of degree n>0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zero ex. f(x)=(x-1)(x^2+2)=(x-1)(x-sqrt(2)i)(x+sqrt(2)i)
   • Principle square root of a negative number: if a is a positive number, then the principle square root of the negative number -a is defined as; sqrt(-a)=sqrt(a)i