Synthetic Division and Remainder Theorem
Synthetic Division is a condensed method of long division. It is quick and easy.
Unfortunately, it can only be used when the divisor is in the form of ( x  a )
Synthetic division:
9 x 2  5x  1
x 1
1.
x3  125
x5
2.
5 x3  2 x  4
x 1
3.
Reminders:
1. Write both polynomials
in standard form.
2. Fill in all missing
terms with a place holder
of zero.
3. Write your answer as a
polynomial that is one
degree less than the
dividend (numerator).
Remainder Theorem:
When dividing a polynomial by ( x  c ) , then the remainder is f ( c) .
3
If f ( x )  5 x  2 x  4 , find f (1) . (refer to #3)
3
2
If f ( x)   x  11x  7 x  19 , find f ( 1) .
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4
3
2
.
3
2
If f ( x )  12 x  x  6 x  5 , find f 
Factor Theorem:
Let f ( x) be a polynomial. Then if f (c)  0 , then ( x  c ) is a factor of f ( x ) . And if
( x  c ) is a factor of f ( x ) , then f (c )  0 . (refer to #2)
3
2
Solve the equation 2 x  3 x  11x  6  0 , given that 3 is a zero (or factor) of the function
f ( x )  2 x3  3 x 2  11x  6
4
3
2
Solve the equation 3 x  17 x  19 x  21x  18  0 , given that 3 is a zero of multiplicity of
4
3
2
two of the function f ( x)  3 x  17 x  19 x  21x  18
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