Difference between revisions of "Factor Theorem"
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− | The '''Factor Theorem''' | + | The '''Factor Theorem''' says that if <math>P(x)</math> is a [[polynomial]], then <math>x-a</math> is a [[factor]] of <math>P(x)</math> [[iff]] <math>P(a)=0</math>. |
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==Proof== | ==Proof== | ||
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Now suppose that <math>P(a) = 0</math>. | Now suppose that <math>P(a) = 0</math>. | ||
− | Apply division [[algorithm]] to get <math>P(x) = (x - a)Q(x) + R(x)</math>, where <math>Q(x)</math> is a polynomial with <math>\deg(Q(x)) = \deg(P(x)) - 1</math> and <math>R(x)</math> is the [[remainder polynomial]] such that <math>0\le\deg(R(x)) < \deg(x - a) = 1</math>. | + | Apply division [[algorithm]] to get <math>P(x) = (x - a)Q(x) + R(x)</math>, where <math>Q(x)</math> is a polynomial with <math>\deg(Q(x)) = \deg(P(x)) - 1</math> and <math>R(x)</math> is the [[remainder polynomial]] such that <math>0\le\deg(R(x)) < \deg(x - a) = 1</math>. This means that <math>R(x)</math> can be at most a [[constant]] polynomial. |
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− | This means that <math>R(x)</math> can be at most a [[constant]] polynomial | ||
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− | + | Substitute <math>x = a</math> and get <math>P(a) = (a - a)Q(a) + R(a) = 0\Rightarrow R(a) = 0</math>. Since <math>R(x)</math> is a constant polynomial, <math>R(x) = 0</math> for all <math>x</math>. | |
Therefore, <math>P(x) = (x - a)Q(x)</math>, which shows that <math>x - a</math> is a factor of <math>P(x)</math>. | Therefore, <math>P(x) = (x - a)Q(x)</math>, which shows that <math>x - a</math> is a factor of <math>P(x)</math>. | ||
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[[Category:Elementary algebra]] | [[Category:Elementary algebra]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 02:22, 21 April 2008
The Factor Theorem says that if is a polynomial, then
is a factor of
iff
.
Proof
If is a factor of
, then
, where
is a polynomial with
. Then
.
Now suppose that .
Apply division algorithm to get , where
is a polynomial with
and
is the remainder polynomial such that
. This means that
can be at most a constant polynomial.
Substitute and get
. Since
is a constant polynomial,
for all
.
Therefore, , which shows that
is a factor of
.
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