1st Year Math Notes Chapter 5

Important Complete 1st Year Math Notes Chapter 5 of Partial Fractions written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Partial Fractions for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.

• Our comprehensive 1st Year Math Notes Chapter 5 will ensure you're fully prepared for your exams.
• Definition of a rational function with examples
• Definition of a proper rational function with examples
• Definition of an improper rational function with examples
• Definition of Partial Fraction with examples
• Definition of Partial Fraction Resolution with examples
• Definition of Irreducible quadratic factor with examples
• Solution examples and exercises Questions
• Note: In the solution of examples 1 and 2. We observe that the value of the constants have been found by substituting those values of x in the identities which can be got by putting each linear factor of the denominators equal to zero.
• Expressing a rational function as a sum of partial fractions is called Partial Fraction Resolution. It is an extremely valuable tool in the study of calculus.
• An open sentence formed by using the sign of equality ‘=’ is called an equation. The equations can be divided into the following two kinds:
• Conditional equation: It is an equation in which two algebraic expressions are equal for particular value/s of the variable e.g.,
• Identity: It is an equation which holds good for all values of the variable e.g.,a) (a + b) x = ax + bx is an identity and its two sides are equal for all values of x.b) (x + 3) (x + 4) = x2+ 7x + 12 is also an identity which is true for all values of x. For convenience, the symbol “=” shall be used both for equation and identity.
• When a rational fraction is separated into partial fractions, the result is an identity; i.e., it is true for all values of the variable. The evaluation of the coeicients of the partial fractions is based on the following theorem: “If two polynomials are equal for all values o f the variable, then the polynomials have same degree and the coeicients of like powers of the variable in both the polynomials must be equal”.
• Resolution of a Rational Fraction P(x) / Q (x) into Partial Fractions following are the main points of resolving a rational fraction P(x) / Q (x) into partial fractions:
• i) The degree of P(x) must be less than that of Q(x). If not, divide and work with the remainder theorem.
• ii) Clear the given equation of fractions.
• iii) Equate the coeicients of like terms (powers of x).
• iv) Solve the resulting equations for the coeicients.
• Deinition: A quadratic, factor is irreducible if it cannot be written as the product of two linear factors with real coeicients. For example, x2+ x + 1 and x2+ 3 are irreducible quadratic factors. 
• If the polynomial Q(x) contains non-repeated irreducible quadratic factor then  P(x) / Q (x) may be written as identity having partial fractions.
• Rational Fraction We know that p / q where p, q U Z and q ≠ 0 is called a rational number. Similarly, the quotient of two polynomial P(x) / Q (x) where Q x( ) 0, ≠ with no common factors, is called a Rational Fraction. A rational fraction is of two types: 
• Proper Rational Fraction: A rational fraction  P(x) / Q (x) is called a Proper Rational Fraction if the degree of the polynomial P(x) in the numerator is less than the degree of the polynomial Q (x) in the denominator.
• Improper Rational Fraction:  A rational fraction P(x) / Q (x) is called an Improper Rational Fraction if the degree of the polynomial P(x) in the numerator is equal to or greater than the degree of the polynomial Q (x) in the denominator.