Partial Fractions
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Questions and Answers

What is the first step when dealing with an improper fraction in partial fraction decomposition?

  • Apply the remainder theorem
  • Express the fraction as a sum of a polynomial and a proper fraction by division (correct)
  • Find the values of x that make the denominator zero
  • Decompose the fraction into partial fractions

    • What is the purpose of partial fraction decomposition?

  • To integrate rational functions (correct)
  • To find the remainder of a polynomial division
  • To simplify algebraic expressions
  • To factorize quadratic expressions

    • 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) (correct)
  • (4/3) / (x-1) + (11/3) / (x+2)
  • (-4/3) / (x+1) + (11/3) / (x-2)
  • (4/3) / (x+2) + (11/3) / (x-1)

    • What is the coefficient of x^n in the expression expanded in ascending order?

    <p>(-2)(-1)^n / (3*2^n)</p> Signup and view all the answers

    What substitution can sometimes help in integrating a rational function by partial fractions?

    <p>Substituting x with its reciprocal</p> Signup and view all the answers

    What is the condition for a rational function f(x) / g(x) to be called a proper rational function?

    <p>The degree of f(x) is less than the degree of g(x).</p> Signup and view all the answers

    What is the process of breaking a rational function into simpler fractions called?

    <p>Decomposition of rational expressions</p> Signup and view all the answers

    What is the condition for a rational function to be broken down into partial fractions?

    <p>The degree of the numerator is less than the degree of the denominator.</p> Signup and view all the answers

    What is the purpose of partial fraction decomposition?

    <p>To break down a rational function into simpler fractions.</p> Signup and view all the answers

    What is an example of an improper rational function?

    <p>x^3 / (x - 1)(x - 2)</p> Signup and view all the answers

    What is the form of the partial fraction corresponding to a non-repeated linear factor (x - a) in the denominator of a proper fraction?

    <p>A / (x - a)</p> Signup and view all the answers

    How do we determine the constants A1, A2, ..., An in the partial fraction decomposition of a rational expression?

    <p>By equating the numerators of the LHS and RHS and substituting x = a1, a2, ..., an</p> Signup and view all the answers

    What is the remainder of a polynomial f(x) when divided by (x - a)?

    <p>f(a)</p> Signup and view all the answers

    What is the partial fraction decomposition of the expression 1 / ((x - a)(x - b))?

    <p>(A / (x - a)) + (B / (x - b))</p> Signup and view all the answers

    What is the remainder of x^64 + x^27 + 1 when divided by (x + 1)?

    <p>1</p> Signup and view all the answers

    What is the correct partial fraction of the expression 3x + 4 / (x + 1)^2 (x - 1)?

    <p>A / (x + 1) + B / (x + 1)^2 + C / (x - 1)</p> Signup and view all the answers

    If the denominator of a rational function consists of a non-repeated quadratic factor ax^2 + bx + c, what is the corresponding partial fraction?

    <p>(Ax + B) / (ax^2 + bx + c)</p> Signup and view all the answers

    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)?

    <p>7 = A(2)^2</p> Signup and view all the answers

    What is the form of the partial fraction corresponding to a repeated linear factor (x - a)^n in the denominator of a proper fraction?

    <p>A / (x - a) + B / (x - a)^2 +... + C / (x - a)^n</p> Signup and view all the answers

    What is the purpose of substitution in partial fraction decomposition?

    <p>To put the equation in a more solvable form</p> Signup and view all the answers

    If 2x + 3 = a(x - 3) + b(x + 1), then what is the value of a + b?

    <p>9/4</p> Signup and view all the answers

    What is the partial fraction decomposition of the expression 3x + 10 / (x - 2)(x - 1)?

    <p>A/(x - 2) + B/(x - 1) + 3</p> Signup and view all the answers

    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)?

    <p>A/(x - a) + B/(x - a)^2 +... + C/(x - a)^k</p> Signup and view all the answers

    What is the condition for a rational function f(x) / g(x) to be broken down into partial fractions?

    <p>All of the above</p> Signup and view all the answers

    If f(x) = A/(x - 1) + B/(x - 2) + C/(x - 3), what is the value of f(4)?

    <p>A/3 + B/2 + C/1</p> Signup and view all the answers

    What is the value of A in the partial fraction decomposition of the expression (3x - 1) / ((x + 2)(x - 1)(x + 1))?

    <p>1</p> Signup and view all the answers

    What is the result of the partial fraction decomposition of the expression x / ((x + 1)^2(x - 2))?

    <p>(Ax + B) / (x + 1) + C / (x + 1)^2 + D / (x - 2)</p> Signup and view all the answers

    What is the value of C in the partial fraction decomposition of the expression (x - 1) / ((x + 2)(x - 1)(x + 1))?

    <p>-1</p> Signup and view all the answers

    What is the form of the partial fraction corresponding to a non-repeated linear factor (x - a) in the denominator of a proper fraction?

    <p>A / (x - a)</p> Signup and view all the answers

    What is the purpose of partial fraction decomposition of a rational function?

    <p>To break down the rational function into simpler fractions</p> Signup and view all the answers

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