If (1)/((a^2-b x)(b^1-a x))=(A)/(a^2-b x)+(B)/(b^2-a x), then the value of A and B respectively

step 1 Hello. So the question is taken from the partial friction where we need to convert this expressi

If (1)/((a^2-b x)(b^1-a x))=(A)/(a^2-b x)+(B)/(b^2-a x),...

If $\frac{1}{\left(a^{2}-b x\right)\left(b^{1}-a x\right)}=\frac{A}{a^{2}-b x}+\frac{B}{b^{2}-a x}$, then the value of $A$ and $B$ respectively would be (1) $\frac{b}{b^{3}-a^{3}}, \frac{a}{b^{3}-a^{3}}$ (2) $\frac{b}{b^{3}-a^{3}}, \frac{a}{a^{3}-b^{3}}$ (3) $\frac{b}{a^{3}-b^{3}}, \frac{a}{a^{3}-b^{3}}$ (4) $\frac{-b}{b^{3}-a^{3}}, \frac{-a}{a^{3}-b^{3}}$

If $\frac{1}{\left(a^{2}-b x\right)\left(b^{1}-a x\right)}=\frac{A}{a^{2}-b x}+\frac{B}{b^{2}-a x}$, then the value of $A$ and $B$ respectively would be
(1) $\frac{b}{b^{3}-a^{3}}, \frac{a}{b^{3}-a^{3}}$
(2) $\frac{b}{b^{3}-a^{3}}, \frac{a}{a^{3}-b^{3}}$
(3) $\frac{b}{a^{3}-b^{3}}, \frac{a}{a^{3}-b^{3}}$
(4) $\frac{-b}{b^{3}-a^{3}}, \frac{-a}{a^{3}-b^{3}}$

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Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This approach is particularly useful when integrating rational functions or simplifying them by breaking down a complex fraction into a series of terms with simpler denominators.

Clearing the Denominator

Clearing the denominator involves multiplying both sides of an equation by the common denominator to eliminate fractions. This step transforms the equation into a polynomial identity, which is easier to work with when solving for unknown constants in the decomposition.

Equating Coefficients

Once the denominators have been cleared, the resulting equation is typically a polynomial identity. Equating the coefficients of like powers of the variable on both sides of the identity is a standard method to set up a system of linear equations, thereby allowing the determination of the unknown constants.

System of Linear Equations

After equating coefficients, a system of linear equations is formed whose solution gives the values of the unknown constants in the partial fraction decomposition. Solving this system is a fundamental algebraic skill, often involving substitution or elimination methods.

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