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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)