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4.1 Notes on Advanced Algebra Techniques
Long Division:
Example 1. Evaluate the following
2 x  5 4 x3  8 x 2  7 x  2
Example 2. Use long division to find the quotient and the remainder when
divided by
2 x2  x  1
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2 x 4  3x3  2 x  1 is
Synthetic Division:
Example 3. Divide the following using synthetic division. Write a summary statement in fraction form.
3x3  2 x 2  6 x  10
x2
Remainder Theorem:
If a polynomial f ( x ) is divided by x  k , then the remainder is r  f (k)
3
Example 4. Find the remainder when 2 x  4 x  1 is divided by
1.
x 1
2.
x 5
3.
x3
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Rational Zeros Theorem:
A. Real zeros of polynomial functions can be either rational or irrational
B. Suppose f is a polynomial function of degree n > 0 of the form an x n  an1 x n1  ...  a0 with
every coefficient an integer and a0  0 . If
p
is a rational zero of f, where p and q have no
q
common integer factors other than 1, then
a. p is an integer factor of the constant coefficient a0
b.
q is an integer factor of the leading coefficient an
*Not everything in this list will be a root, and non-integer roots won’t be in the list, but if we are
lucky, we will be able to identify a root with this method. Test to see if these equal zero, then use
synthetic to find the rest.)
Example 5. List all possible zeros of the following functions.
a.
g ( x)  3 x 3  2 x  4
b.
h( x)  6 x 4  2 x3  x  5
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Example 6. List all rational zeros of the following functions. (Find one using the rational roots theorem,
then use synthetic!)
a.
f ( x)  2 x3  x 2  13x  6
b.
f ( x)  x 3  4 x 2  1
Example 7. Write the equation in standard form for the quartic polynomial with zeros 1, -1, 2, -1/2
through the point (-2,9).
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