Remainder and Factor Theorems

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Remainder and Factor Theorems

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

Find the remainder when is divided by Find f(4) for the given function

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

π

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)

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

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; (a+bi) and (a-bi)

Examples: Find ALL the zeros

Write the polynomial with the given solutions:

Write the polynomial: