Answer:
(x³ + x² + x + 2) / (x² - 1)
Using long division, divide the two expressions.
(x + 1) + ((2x + 3) / (x² - 1))
Using a² - b² = (a + b)(a - b), factor:
(x + 1) + ((2x + 3) / (x + 1)(x - 1))
Isolate the fraction.
((2x + 3) / (x + 1)(x - 1))
For each factor in the denominator, make new fractions using the factors as new denominators while the numerators are unknown values.
((2x + 3) / (x + 1)(x - 1)) = (? / x - 1) + (? / x + 1)
Set constants for each of the unknown values.
((2x + 3) / (x + 1)(x - 1) = (A / x - 1) + (B / x + 1)
Multiply both sides by (x + 1)(x - 1).
2x + 3 = (x + 1)A + (x - 1)B
Simplify.
2x + 3 = Ax + A + Bx - B
Group the terms.
2x + 3 = (A + B)x + (A - B)
When the two polynomials are equal, their corresponding coefficients must also be equal.
3 = A + B
2 = A - B
Solve the system of equations.
(A, B) = (5/2, -½)
Substitute the given values into the new fraction.
(5/2 / x - 1) + (-½ / x + 1)
Simplify the expression.
(5 / 2(x - 1)) - (1 / 2(x + 1))
Return the partial-fraction decomposition into the expression.
= x + 1 + (5/ 2(x - 1)) - (1/ 2(x + 1))
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Answers & Comments
Answer:
(x³ + x² + x + 2) / (x² - 1)
Using long division, divide the two expressions.
(x + 1) + ((2x + 3) / (x² - 1))
Using a² - b² = (a + b)(a - b), factor:
(x + 1) + ((2x + 3) / (x + 1)(x - 1))
Isolate the fraction.
((2x + 3) / (x + 1)(x - 1))
For each factor in the denominator, make new fractions using the factors as new denominators while the numerators are unknown values.
((2x + 3) / (x + 1)(x - 1)) = (? / x - 1) + (? / x + 1)
Set constants for each of the unknown values.
((2x + 3) / (x + 1)(x - 1) = (A / x - 1) + (B / x + 1)
Multiply both sides by (x + 1)(x - 1).
2x + 3 = (x + 1)A + (x - 1)B
Simplify.
2x + 3 = Ax + A + Bx - B
Group the terms.
2x + 3 = (A + B)x + (A - B)
When the two polynomials are equal, their corresponding coefficients must also be equal.
3 = A + B
2 = A - B
Solve the system of equations.
(A, B) = (5/2, -½)
Substitute the given values into the new fraction.
(5/2 / x - 1) + (-½ / x + 1)
Simplify the expression.
(5 / 2(x - 1)) - (1 / 2(x + 1))
Return the partial-fraction decomposition into the expression.
= x + 1 + (5/ 2(x - 1)) - (1/ 2(x + 1))