# Remainder Factor and Partial fractions ```Aim: Be able to analyse and model
engineering situations and solve
problems using algebraic methods
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
1
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Polynomial division is done as long division.
Factor Theorem states that if x= a is a root of
the equation f(x) = 0, then x-a is a factor of f(x).
Thus, a factor of x-a corresponds to a root of
x=a
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
2
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It is of importance to know the remainder of a
polynomial division because if the remainder is
zero, it means that that divisor is a factor of the
polynomial, and therefore , crucial in factorising
that particular polynomial.
The Remainder Theorem states: If (ax2 +bx +c) is
divided by (x-p), the remainder will be (ap2 + bp +
c)
This is true for cubic and higher order equations.
So we write:
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
3
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If (ax3 +bx2 +cx + d) is divided by (x-p), the
remainder will be (ap3 + bp2 + cp + d)
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
4
Partial Fractions
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Partial fractions are a reverse process of polynomial division to
convert them into fractions of its factors.
In order to work out the solution of this reverse process into
partial fractions:
The denominator must factorise.
The numerator must be at least one degree less than the
denominator.
When the degree of the numerator is equal to or higher than
denominator, the numerator must be divided by the denominator
by long division.
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
5
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There are three types of partial fractions with
denominators containing:
Linear factors
Repeated linear factors
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
6
Linear Types
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Expression:
f(x) /(x+a)(x-b)(x+c)
Partial fraction form:
A/(x+a) + B/(x-b) + C/(x+c)
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
7
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f(x)/(x+a)3
Form of partial fractions:
A/(x+a) + B/(x+a)2 + C/(x+a)3
For Quadratic factors, form of partial fractions:
f(x)/(ax2 + bx + c)(x+d) =
(Ax +B)/(ax2 + bx + c)(x+d) + C/(x+d)
14/08/2022
Louis Mbua Egbe PhD Teach
Engineering &amp; Technology Solutions
Ltd &copy;
8
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