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Factoring Polynomials of Higher Degree
The Factor Theorem Part II
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Factoring Polynomials of Higher Degree
From last class, synthetic division is a
quick process to divide polynomials by
binomials of the form x – a and bx – a.
2 1 -4 -7 10
1
2
-2
-4
-11 -12
-22
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Factoring Polynomials of Higher Degree
Example 1: Use synthetic division to find the
quotient and remainder when 6x3 – 25x2 – 29x + 25
is divided by 2x – 1.
Solution:
b
a
6 -25 -29 25
6
3
-22
-11
-40 5
-20bx – a is written
in the form in
front of the “L”.
1
2
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Factoring Polynomials of Higher Degree
The Factor Theorem (Part 2)
If p(x) = anxn + an-1xn-1 + an-2xn-2 +…+ axx2 + azx + a0 is a
polynomial function with integer coefficients, and if bx – a
is a factor of p(x) where “b” and “a” are also integers, then
“b” is a factor of the leading coefficient an and “a” is a
factor of the constant term a0.
“b” and “a” should have no factors in common.
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Factoring Polynomials of Higher Degree
Example 2: Factor x3 + 5x2 + 2x – 8.
Solution: If x3 + 5x2 + 2x – 8 has a factor of
the form bx – a, then “a” is a factor of -8 and
“b” is a factor of 1.
A list of possible factors is x – 1, x – 2, x – 4,
x – 8, x + 1, x + 2, x + 4, x + 8.
We will use synthetic division to try and find one factor. Start by testing x – 1.
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Factoring Polynomials of Higher Degree
1 5 2 -8
1
1
6
6
8 0
8
1
We note that x – 1 must be a factor since the remainder is 0.
Thus, we have:
x3 + 5x2 + 2x – 8 = (x – 1)(x2 + 6x + 8)
= (x – 1)(x + 2)(x +4)
Factor the trinomial
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Factoring Polynomials of Higher Degree
Note that only 3 of the possible 8 solutions for the polynomial were factors.
It is unlikely that you will actually find a factor on your first try.
It will take a few attempts to find the first one, and then factor the remaining quadratic by inspection.
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SummaryTo factor a polynomial of higher degree:
1) Write the polynomial in decreasing degree inserting 0’s for any missing terms.
2) Make a list of all possible factors bx – a, where “a” is a factor of the constant term and “b” is a factor of the leading coefficent. “b” and “a” should have no factors in common.
3) Test the list of potential factors using synthetic division.
4) When you find a factor, factor the remaining quadratic.
Factoring Polynomials of Higher Degree
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Homework
Do # 19, 25, 29, 31, 33, and 35 on page 130 from section 4.3 for Monday April 20th
Have a Safe and Fun Easter Break