Advertisements
Advertisements
प्रश्न
Resolve the following rational expressions into partial fractions
`(x^3 + 2x + 1)/(x^2 + 5x + 6)`
उत्तर
`(x^3 + 2x + 1)/(x^2 + 5x + 6)`
Since the numerator is of degree greater than that of the denominator divide the numerator by the denominator.
∴ `(x^3 + 2x + 1)/(x^2 + 5x + 6) = (x - 5) + (21x + 31)/(x^2 + 5x + 6)` ......(1)
Consider `(21x + 31)/(x^2 + 5x + 6)`
`(21x + 31)/(x^2 + 5x + 6) = (21x + 31)/(x^2 + 3x + 2x + 6)`
= `(21x + 31)/(x(x + 3) + 2(x + 3))`
= `(21x + 31)/((x + 2)(x + 3))`
`(21x + 31)/(x^2 + 5x + 6) = "A"/(x + 2) + "B"/(x + 3)` ......(2)
`(21x + 31)/(x^2 + 5x + 6) = ("A"(x + 3) + "B"(x + 2))/((x + 2)(x + 3))`
21x + 31 = A(x + 3) + B(x + 2) ......(3)
Put x = – 3, in equation (3)
21(– 3) + 31 = A(– 3 + 3) + B(– 3 + 2)
– 63 + 31 = 0 – B
– 32 = – B
⇒ B = 32
Put x = – 2, in equation (3)
21(– 2) + 31 = A(– 2 + 3) + B(– 2 + 2)
– 42 + 31 = A + 0
⇒ A = – 11
Substituting the values of A and B in equation (2)
We have `(21x + 31)/(x^2 + 5x + 6) = (-11)/(x + 2) + 32/(x + 3)`
= `- 11/(x + 2) + 32/(x + 3)`
∴ The required partial fraction is
`(x^3 + 2x + 1)/(x^2 + 5x + 6) = (x - 5) - 11/(x + 2) + 32/(x + 3)`
APPEARS IN
संबंधित प्रश्न
Find all values of x for which `(x^3(x - 1))/((x - 2)) > 0`
Find all values of x that satisfies the inequality `(2x - 3)/((x - 2)(x - 4)) < 0`
Resolve the following rational expressions into partial fractions
`x/((x^2 + 1)(x - 1)(x + 2))`
Resolve the following rational expressions into partial fractions
`x/((x - 1)^3`
Resolve the following rational expressions into partial fractions
`1/(x^4 - 1)`
Resolve the following rational expressions into partial fractions
`(x - 1)^2/(x^3 + x)`
Resolve the following rational expressions into partial fractions
`(x^2 + x + 1)/(x^2 - 5x + 6)`
Resolve the following rational expressions into partial fractions
`(x + 12)/((x + 1)^2 (x - 2))`
Resolve the following rational expressions into partial fractions
`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`
Determine the region in the plane determined by the inequalities:
x ≤ 3y, x ≥ y
Determine the region in the plane determined by the inequalities:
3x + 5y ≥ 45, x ≥ 0, y ≥ 0
Determine the region in the plane determined by the inequalities:
2x + 3y ≤ 35, y ≥ 2, x ≥ 5.
Determine the region in the plane determined by the inequalities:
2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0
Determine the region in the plane determined by the inequalities:
x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0
Choose the correct alternative:
The solution of 5x − 1 < 24 and 5x + 1 > −24 is
Choose the correct alternative:
The solution set of the following inequality |x − 1| ≥ |x − 3| is
