30 Questions
What is the first step when dealing with an improper fraction in partial fraction decomposition?
Express the fraction as a sum of a polynomial and a proper fraction by division
What is the purpose of partial fraction decomposition?
To integrate rational functions
What is the result of the partial fraction decomposition of the expression 5x+6 / (2+x)(1-x)?
(-4/3) / (x-2) + (11/3) / (x+1)
What is the coefficient of x^n in the expression expanded in ascending order?
(-2)(-1)^n / (3*2^n)
What substitution can sometimes help in integrating a rational function by partial fractions?
Substituting x with its reciprocal
What is the condition for a rational function f(x) / g(x) to be called a proper rational function?
The degree of f(x) is less than the degree of g(x).
What is the process of breaking a rational function into simpler fractions called?
Decomposition of rational expressions
What is the condition for a rational function to be broken down into partial fractions?
The degree of the numerator is less than the degree of the denominator.
What is the purpose of partial fraction decomposition?
To break down a rational function into simpler fractions.
What is an example of an improper rational function?
x^3 / (x - 1)(x - 2)
What is the form of the partial fraction corresponding to a non-repeated linear factor (x - a) in the denominator of a proper fraction?
A / (x - a)
How do we determine the constants A1, A2, ..., An in the partial fraction decomposition of a rational expression?
By equating the numerators of the LHS and RHS and substituting x = a1, a2, ..., an
What is the remainder of a polynomial f(x) when divided by (x - a)?
f(a)
What is the partial fraction decomposition of the expression 1 / ((x - a)(x - b))?
(A / (x - a)) + (B / (x - b))
What is the remainder of x^64 + x^27 + 1 when divided by (x + 1)?
1
What is the correct partial fraction of the expression 3x + 4 / (x + 1)^2 (x - 1)?
A / (x + 1) + B / (x + 1)^2 + C / (x - 1)
If the denominator of a rational function consists of a non-repeated quadratic factor ax^2 + bx + c, what is the corresponding partial fraction?
(Ax + B) / (ax^2 + bx + c)
What is the result of putting x = 1 in the equation 3x + 4 = A(x + 1)^2 + B(x + 1)(x - 1) + C(x - 1)?
7 = A(2)^2
What is the form of the partial fraction corresponding to a repeated linear factor (x - a)^n in the denominator of a proper fraction?
A / (x - a) + B / (x - a)^2 +... + C / (x - a)^n
What is the purpose of substitution in partial fraction decomposition?
To put the equation in a more solvable form
If 2x + 3 = a(x - 3) + b(x + 1), then what is the value of a + b?
9/4
What is the partial fraction decomposition of the expression 3x + 10 / (x - 2)(x - 1)?
A/(x - 2) + B/(x - 1) + 3
If g(x) = (x - a)^k (x - a1)(x - a2)......(x - ar), what is the form of the partial fraction corresponding to the linear factor (x - a)?
A/(x - a) + B/(x - a)^2 +... + C/(x - a)^k
What is the condition for a rational function f(x) / g(x) to be broken down into partial fractions?
All of the above
If f(x) = A/(x - 1) + B/(x - 2) + C/(x - 3), what is the value of f(4)?
A/3 + B/2 + C/1
What is the value of A in the partial fraction decomposition of the expression (3x - 1) / ((x + 2)(x - 1)(x + 1))?
1
What is the result of the partial fraction decomposition of the expression x / ((x + 1)^2(x - 2))?
(Ax + B) / (x + 1) + C / (x + 1)^2 + D / (x - 2)
What is the value of C in the partial fraction decomposition of the expression (x - 1) / ((x + 2)(x - 1)(x + 1))?
-1
What is the form of the partial fraction corresponding to a non-repeated linear factor (x - a) in the denominator of a proper fraction?
A / (x - a)
What is the purpose of partial fraction decomposition of a rational function?
To break down the rational function into simpler fractions
Learn about partial fractions and how to apply them to integrate rational functions. Understand the importance of proper fractions and suitable substitutions. Practice with examples to master the technique.
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