Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 7 Partial Fractions to solve questions creatively.

## Intermediate 2nd Year Maths 2A Partial Fractions Formulas

→ The quotient of two polynomials f(x) and Φ(x) where Φ(x) ≠ 0, is called a rational fraction.

→ If the degree of f(x) < the degree of Φ(x) in a rational fraction \(\frac{f(x)}{\phi(x)}\), then the rational fraction is called a proper fraction.

→ If the degree of f(x) ≥ the degree of Φ(x) in a rational fraction \(\frac{f(x)}{\phi(x)}\), then the rational fraction is called a proper fraction.

→ Let \(\frac{f(x)}{\phi(x)}\) be a proper fraction

→ When Φ(x) contains non-repeated linear factors only corresponding to every non-repeated linear factor (ax +b) of Φ(x) there exists a partial fraction of the form \(\frac{A}{a x+b}\) where A is a real number.

→ When Φ(x) contains repeated and non-repeated linear factors only corresponding to every repeated linear factor (ax + b)^{p} of Φ(x) there exist fractions of the form.

\(\frac{A_{1}}{a x+b}+\frac{A_{2}}{(a x+b)^{2}}+\ldots+\frac{A_{p}}{(a x+b)^{p}}\) Where A_{1}, A_{2}, A_{3} ………… A_{p} are real numbers.

→ When Φ(x) contains non-repeated irreducible factors only. Corresponding to every non-repeated irreducible quadratic factor ax^{2} + bx + c of Φ(x) of exists a partial fraction of the form \(\frac{A x+B}{a x^{2}+b x+C}\) where A and B are real numbers.

→ When Φ(x) contains repeated and non-repeated irreducible factors only. Corresponding to every repeated quadratic factor (ax^{2} + bx + c0^{p} of Φ(x) there exists the partial fractions of the form

\(\frac{A_{1} x+B_{1}}{a x^{2}+b x+c}\) + \(\frac{A_{2} x+B_{2}}{\left(a x^{2}+b x+c\right)^{2}}\) + ……… + \(\frac{A_{p} x+\dot{B}_{p}}{\left(a x^{2}+b x+c\right)^{p}}\) where A_{1}, A_{2}, …….. A^{p} and B_{1}, B_{2}, ………. B_{p} are real numbers.

→ Let \(\frac{f(x)}{\phi(x)}\) be a improper fraction, then

\(\frac{f(x)}{\phi(x)}\) = Q(x) + \(\frac{R(x)}{\phi(x)}\) where Q(x) is quotient and R(x) is the remainder, and Degree R(x) < that of Φ(x).

→ Remainder obtained when f(x) is divided by x – a is f(a).

If degree of divisor is ‘n’, then the degree of remainder is (n – 1)

f(x), g(x) are two polynomials. If g(x) ≠ 0, then ∃ two polynomials q(x) , r(x) such that

\(\frac{f(x)}{g(x)}\) = q(x) + \(\frac{r(x)}{g(x)}\) if the degree of f(x) is > that of g(x)

II. Method of resolving proper fraction \(\frac{f(x)}{g(x)}\) into partial fractions.

Type 1 : When the denominator g(x) contains non-repeated factors i.e

g(x) = (x – a)(x – b)(x – c)

\(\frac{f(x)}{(x-a)(x-b)(x-c)}=\frac{\mathrm{A}}{\mathrm{x}-\mathrm{a}}+\frac{\mathrm{B}}{\mathrm{x}-\mathrm{b}}+\frac{\mathrm{C}}{\mathrm{x}-\mathrm{c}}\)

Type 2 : When the denominator g(x) contains repeated and non repeated linear factors

i.e g(x) = (x – a)^{2}(x – b)

\(\frac{f(x)}{(x-a)^{2}(x-b)}=\frac{A}{x-a}+\frac{B}{(x-a)^{2}}+\frac{C}{(x-b)}\)

Type 3 : When the denominator g(x) contains non repeated irreducible quadratic factors

i.e g(x) = (ax^{2} + bx + c)(x – d)

Type 4 : When the denominator g(x) contains repeated irreducible quadratic factors

i.e g(x) = (ax^{2} + bx + c)^{2}(x – d)

\(\frac{f(\mathrm{x})}{\left(\mathrm{ax}{ }^{2}+\mathrm{bx}+\mathrm{c}\right)^{2}(\mathrm{x}-\mathrm{d})}=\frac{\mathrm{Ax}+\mathrm{B}}{\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)}+\frac{\mathrm{Cx}+\mathrm{D}}{\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)^{2}}+\frac{\mathrm{E}}{\mathrm{x}-\mathrm{d}}\)