NON REPEATED LINEAR FACTORS - PUNJAB-CURRICULUM-AND-TEXTBOOK-BOARD-PCTB / 10TH-CLASS / MATHEMATICS / MATHEMATICS-10-10TH-CLASS-NBF-2024 / PARTIAL-FRACTIONS / NON-REPEATED-LINEAR-FACTORS

4.2.1 Resolution of an algebraic fraction into partial fractions, when $D(x)$ consists of non- repeated linear factors.

Rule I: If linear factor $(ax + b)$ occurs as a factor of $D(x)$ , then there is a partial fraction of the form $\frac{A}{a x + b}$ , where $A$ is a constant to be found.

In $\frac{N (x)}{D (x)}$ , the polynomial $D(x)$ may be written as,

D(x) = (a_{1}x + b_{1})(a_{2}x + b_{2})\dots (a_{n}x + b_{n})

We have, $\frac{N (x)}{D (x)} = \frac{A_{1}}{a_{1} x + b_{1}} + \frac{A_{2}}{a_{2} x + b_{2}} + \frac{A_{3}}{a_{3} x + b_{3}} + \dots + \frac{A_{n}}{a_{n} x + b_{n}},$

where $A_{1}A_{2}\dots A_{n}$ are constants to be determined. The following examples illustrate how we can find these constants:

Example 1:

Resolve $\frac{5 x + 4}{(x - 4) (x + 2)}$ into partial fractions.

Solution: Let $\frac{5 x + 4}{(x - 4) (x + 2)} = \frac{A}{x - 4} + \frac{B}{x + 2}$ (i)

Multiplying throughout by $(x - 4)(x + 2)$ , we get

5x + 4 = A(x + 2) + B(x - 4) \quad (ii)

Equation (ii) is an identity, which holds good for all values of $x$ and hence for

$x = 4 a n d x = - 2.$

Put $x - 4 = 0$ i.e., $x = 4$ (factor corresponding to $A$ ) on both sides of the equation (ii),

we get $5 (4) + 4 = A (4 + 2) \Rightarrow A = 4$

Put $x + 2 = 0$ i.e., $x = - 2$ (factor corresponding to $B$ ), we get

5 (- 2) + 4 = B (- 2 - 4) \Rightarrow - 6 B = - 6 \Rightarrow B = 1

Thus required partial fractions are $\frac{4}{x - 4} + \frac{1}{x + 2}$

Hence, $\frac{5 x + 4}{(x - 4) (x + 2)} = \frac{4}{x - 4} + \frac{1}{x + 2}$

This method is called the zero's method. This method is especially useful with linear factors in the denominator $D(x)$ .

4.2.1 Non-Repeated Linear Factors

4.2.1 Resolution of an algebraic fraction into partial fractions, when D(x)D(x)D(x) consists of non- repeated linear factors.

Example 1:

4.2.1 Resolution of an algebraic fraction into partial fractions, when $D(x)$ consists of non- repeated linear factors.