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Example Question #31 : Polynomials
If the polynomial
is divided by
,
what is the remainder?
By the Remainder Theorem, if a polynomial is divided by a binomial
, the remainder is
.
Let . Setting
, if
is divided by
, the remainder is
, which can be evaluated by setting
in the definition of
and evaluating:
Example Question #5 : Factoring Polynomials
Which of the following is a factor of the polynomial ?
Call
By the Rational Zeroes Theorem, since has only integer coefficients, any rational solution of
must be a factor of 54 divided by a factor of 1 - positive or negative. 54 has as its factors 1, 2, 3, 6, 9, 18, 27 , 54; 1 has only itself as a factor. Therefore, the rational solutions of
must be chosen from this set:
.
By the Factor Theorem, a polynomial is divisible by
if and only if
- that is, if
is a zero. By the preceding result, we can immediately eliminate
and
as factors, since 12 and 16 have been eliminated as possible zeroes.
Of the three remaining choices, we can demonstrate that is the factor by evaluating
:
, so
is a factor.
Of the remaining two choices, and
, both can be proved to not be factors by showing that
and
are both nonzero:
, so
is not a factor.
, so
is not a factor.
Example Question #7 : How To Factor A Polynomial
If the polynomial
is divided by
,
what is the remainder?
By the Remainder Theorem, if a polynomial is divided by a binomial
, the remainder is
.
Let . Setting
(since
), if
is divided by
, the remainder is
, which can be evaluated by setting
in the definition of
and evaluating:
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