Remainder & Factor Theorems

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Remainder and Factor Theorems
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Remainder Theorem

• If a polynomial f(x) is divided by x a, the
  remainder is the same as finding f(a)

divisor
dividend
quotient
remainder
Synthetic substitution using synthetic division
to evaluate a function
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Find the remainder when
is divided by

Find f(4) for the given function
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Factor Theorem

• x-a is a factor of f(x) if the remainder is zero
• Use this theorem to factor polynomial functions
• Do synthetic division with the given factor and
  then factor the polynomial

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Zeros, Roots and Factors

• c is a zero of f(x)
• x-c is a factor of f(x)
• c is a root or solution of f(x)0
• If c is a real number then (c, 0) is an intercept
  of the graph of f(x)

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Fundamental Theorem of Algebra

• Every polynomial equation with degree greater
  than zero has at least one root in the set of
  complex numbers
• A polynomial of degree n has exactly n roots in
  the set of complex numbers

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To find ALL the zeros

• Use your calculator to find any integer solutions
  in the table
• Use the integer solutions to perform synthetic
  division and reduce the polynomial down to one
  that can be solved by factoring or completing the
  square.
• All imaginary solutions must be in conjugate
  form (abi) and (a-bi)

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Examples Find ALL the zeros
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Write the polynomial with the given solutions
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Write the polynomial