DBM 10013
ENGINEERING
MATHEMATICS 1
Title Layout
Subtitle
1.0 BASIC ALGEBRA
Learning outcomes:
At the end of this topic student should be able to:
❑Partial Fraction
✓ Partial Fraction, Proper and Improper Fractions
✓ Long Division of Polynomials
✓ Linear Factors
✓ Repeated Linear Factors
✓ Improper Partial Fraction
1.3 Partial Fraction
1.3.1 Partial Fraction, Proper and Improper Fractions
✓ Define proper and improper fractions.
✓ Define partial fraction
✓ Convert improper fraction to mixed number by using long division.
✓ Construct partial fraction using proper fraction with:
o c. Linear Factor d.
o Repeated Linear Factor
o Quadratic Factors
QR CODE :Link for Factorization topic
1.3.1 Partial Fraction, Proper and Improper Fractions
So, in order to get the “questions” back from the answer, we must do PARTIAL FRACTION.
Proper Fractions: Improper Fractions:
When degree of numerator is less than degree of When degree of numerator is greather than degree
denominator of denominator
x2 +1 Degree of numerator is 2 ( 2) x(x2 − 25) Degree of numerator is 3 ( 3)
x(x2 − 25) Degree of denomirator is 3 ( 3)
x2 +1 Degree of denomirator is 2 ( 2)
1.3.2 Linear Factors
( ax + b ) (cx + d ) = A b + B d
ax + cx +
1.3.2 Linear Factors
1.3.2 Linear Factors
Example 1
Express the followings as partial fractions.
x+3
( x −1)( x − 2)
Example 1
Example 1
Example 2
Express the followings as partial fractions.
2x −1
x ( x −1)( x + 3)
Example 2
Example 2
Exercise 1
Express the followings as partial fractions.
5
x2 + x − 6
Exercise 2
Express the followings as partial fractions.
2x +3
2x2 + x
1.3.2 Repeated Linear Factors
(ax + b)3 = A + B + c
(ax + b)1 (cx + b)2 (ax + b)3
Example 1
Express the followings as partial fractions.
2x +3
( x −1)2 (x)
Example 2
Express the followings as partial fractions.
x2
(x2 + 2x +1)(x − 3)
Exercise 1
Express the followings as partial fractions.
x2 + 5
(x + 2)(x2 + 2x)
1.3.3 Quadratic Factors
( ) = Ax + B + C
ax2 + b (cx + d ) (ax2 + b) (cx + d )
Example 1
Express the followings as partial fractions.
x −1
(x)( x2 +1)
Example 2
Express the followings as partial fractions.
2
( x2 + 3x +1)( x + 2)
Exercise 1
Express the followings as partial fractions.
x2 + 5
( x2 +1)( x)2
1.3.4 Improper Partial Fractions
1.3.2 Long Division Of Polynomials
Step 5: Bring down 21 to form a new dividend Step 6:Find the second term of the quotient by
devide 7x = 7 +
x
x + 3 x2 +10x + 21
x + 3 x2 +10x + 21
(−)x2 + 3x
7x + 21 (−)x2 + 3x
7x + 21
Step 7 :Multiply (x+3) by 7
Step 8: Substract
+
+
x + 3 x2 +10x + 21
x + 3 x2 +10x + 21
(−)x2 + 3x
7x + 21 (−)x2 + 3x
7x + 21
7x + 21
(−) 7x + 21
0
Example 1
Express the followings as partial fractions.
x2 +1
(x2 −1)