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# remainder theoram.docx

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Department
Mathematics
Course Code
MATA30H3
Professor
Ken Butler

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Description
Remainder Theorem  Long Division can be used to divide a polynomial by a binomial.  The result of the division of a polynomial function P(x) by a binomial of the form x – b can be written as P(x) = (x-b)Q(x) + R where Q(x) is the quotient and R is the remainder.  Division Statement: divisor x quotient + remainder = dividend  can be used to check the result of a division  Remainder theorem states that when a polynomial function P(x) is divided by x – b, the remainder is P(b), and when it is divided by ax-b, the remainder is P(b/a), where a and b are integers and a not = 0. Factor Theorem  For integer values of a and b, with a not equal 0,  Factor Theorem states that x – b is a factor of a polynomial P(x) if and only if P(b) = 0.  Similarly, if ax – b is a factor of P(x) if and only if P(b/a) = 0  Integral Zero Theorem states that if x – b is a factor of a polynomial function P(x) with leading coefficient 1 and remaining coefficients that are integers, then b is a factor of the constant term P(x).  Rational Zero Theorem states that if P(x) is a polynomial function with integer coefficients and x = b/a is a rational zero of P(x), then  b is a factor of the constant term of P(x)  a is a factor of the leading coefficient of P(x)  ax – b is a factor of P(x) Polynomial Equations  Real roots of a polynomial equation P(x) = 0 correspond to the x-intercepts of the graph of the polynomial function P(x).  X-intercepts of the graph
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