Hard A-Level Algebraic and partial fractions Questions

Challenging, exam-style A-Level Algebraic and partial fractions questions with worked solutions. Stretch yourself on the hardest partial-fractions, constants, form problems.

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A-Level34 questionsStep-by-step solutions
Question 1
8 markschallenging
Which of the following is the correct partial fraction decomposition of 2x+3x3−3x−2\frac{2 x + 3}{x^{3} - 3 x - 2}?
Show worked solution

Worked solution

  1. Write the fraction to be decomposed

    2x+3x3−3x−2\frac{2 x + 3}{x^{3} - 3 x - 2}

    We start from the given rational function.

  2. Factorise the denominator

    x3−3x−2=(x−2)(x+1)2x^{3} - 3 x - 2=\left(x - 2\right) \left(x + 1\right)^{2}

    Factorising shows which linear factors appear.

  3. Write the partial fraction form with unknown constants

    2x+3(x−2)(x+1)2=Ax+1+B(x+1)2+Cx−2\frac{2 x + 3}{\left(x - 2\right) \left(x + 1\right)^{2}}=\frac{A}{x + 1}+\frac{B}{\left(x + 1\right)^{2}}+\frac{C}{x - 2}

    Each distinct factor (and each power of a repeated factor) needs its own term.

  4. Cover up \left(x + 1\right)^{2} and substitute x=-1

    1=B (−3) ⇒ B=−131=B\,(-3)\ \Rightarrow\ B=- \frac{1}{3}

    Multiplying by the factor and substituting its root isolates one constant.

  5. Cover up x - 2 and substitute x=2

    7=C (9) ⇒ C=797=C\,(9)\ \Rightarrow\ C=\frac{7}{9}

    Multiplying by the factor and substituting its root isolates one constant.

  6. Substitute x=0 to find A

    x=0: A=−79x=0:\ A=- \frac{7}{9}

    With the other constants known, any convenient value gives the last one.

  7. Multiply through by the denominator to form an identity

    2x+3=A(x2−x−2)+B(x−2)+C(x2+2x+1)2 x + 3=A \left(x^{2} - x - 2\right) + B \left(x - 2\right) + C \left(x^{2} + 2 x + 1\right)

    Clearing the fractions gives an identity true for all x.

  8. Expand the right-hand side

    2x+3=Ax2−Ax−2A+Bx−2B+Cx2+2Cx+C2 x + 3=A x^{2} - A x - 2 A + B x - 2 B + C x^{2} + 2 C x + C

    Expanding lets us compare like terms.

  9. Compare coefficients of like powers of x

    A+C=0, −A+B+2C=2, −2A−2B+C=3A + C=0,\ - A + B + 2 C=2,\ - 2 A - 2 B + C=3

    Matching coefficients confirms the values of the constants.

  10. Express the assumed form over a common denominator

    Ax+1+B(x+1)2+Cx−2=A(x2−x−2)+B(x−2)+C(x2+2x+1)x3−3x−2\frac{A}{x + 1}+\frac{B}{\left(x + 1\right)^{2}}+\frac{C}{x - 2}=\frac{A \left(x^{2} - x - 2\right) + B \left(x - 2\right) + C \left(x^{2} + 2 x + 1\right)}{x^{3} - 3 x - 2}

    Recombining shows the numerators must match.

  11. Verify by recombining the partial fractions

    −79(x+1)−13(x+1)2+79(x−2)=2x+3(x−2)(x+1)2- \frac{7}{9 \left(x + 1\right)} - \frac{1}{3 \left(x + 1\right)^{2}} + \frac{7}{9 \left(x - 2\right)}=\frac{2 x + 3}{\left(x - 2\right) \left(x + 1\right)^{2}}

    Adding the fractions back returns the original expression.

  12. Check the result at a test value

    x=0: −32=−32x=0:\ - \frac{3}{2}=- \frac{3}{2}

    Both sides agree, confirming the decomposition.

  13. Note the degrees involved

    deg⁡N=1, deg⁡D=3\deg N=1,\ \deg D=3

    The degree comparison determines whether division is needed.

  14. Recall the cover-up method

    Multiply by a factor, then substitute its root.\text{Multiply by a factor, then substitute its root.}

    The cover-up method is the quickest route to each constant.

  15. Select the correct decomposition

    2x+3(x−2)(x+1)2=−79(x+1)−13(x+1)2+79(x−2)\frac{2 x + 3}{\left(x - 2\right) \left(x + 1\right)^{2}}=- \frac{7}{9 \left(x + 1\right)} - \frac{1}{3 \left(x + 1\right)^{2}} + \frac{7}{9 \left(x - 2\right)}

    This is the fully correct partial fraction decomposition.

Answer
−79(x+1)−13(x+1)2+79(x−2)- \frac{7}{9 \left(x + 1\right)} - \frac{1}{3 \left(x + 1\right)^{2}} + \frac{7}{9 \left(x - 2\right)}
Question 2
8 markschallenging
Which of the following is the correct partial fraction decomposition of x3+1x2−1\frac{x^{3} + 1}{x^{2} - 1}?
Show worked solution

Worked solution

  1. Write the fraction to be decomposed

    x3+1x2−1\frac{x^{3} + 1}{x^{2} - 1}

    We start from the given rational function.

  2. Compare the degrees of numerator and denominator

    deg⁡N=3 ≥ deg⁡D=2\deg N=3\ \ge\ \deg D=2

    The fraction is improper, so we must divide first.

  3. Divide the numerator by the denominator

    x3+1x2−1=x+x+1x2−1\frac{x^{3} + 1}{x^{2} - 1}=x+\frac{x + 1}{x^{2} - 1}

    Polynomial division gives a polynomial part plus a proper fraction.

  4. Factorise the denominator

    x2−1=(x−1)(x+1)x^{2} - 1=\left(x - 1\right) \left(x + 1\right)

    Factorising shows which linear factors appear.

  5. Write the partial fraction form with unknown constants

    x3+1(x−1)(x+1)=x+Ax−1+Bx+1\frac{x^{3} + 1}{\left(x - 1\right) \left(x + 1\right)}=x+\frac{A}{x - 1}+\frac{B}{x + 1}

    Each distinct factor (and each power of a repeated factor) needs its own term.

  6. Cover up x - 1 and substitute x=1

    2=A (2) ⇒ A=12=A\,(2)\ \Rightarrow\ A=1

    Multiplying by the factor and substituting its root isolates one constant.

  7. Cover up x + 1 and substitute x=-1

    0=B (−2) ⇒ B=00=B\,(-2)\ \Rightarrow\ B=0

    Multiplying by the factor and substituting its root isolates one constant.

  8. Multiply through by the denominator to form an identity

    x+1=A(x+1)+B(x−1)x + 1=A \left(x + 1\right) + B \left(x - 1\right)

    Clearing the fractions gives an identity true for all x.

  9. Expand the right-hand side

    x+1=Ax+A+Bx−Bx + 1=A x + A + B x - B

    Expanding lets us compare like terms.

  10. Compare coefficients of like powers of x

    A+B=1, A−B=1A + B=1,\ A - B=1

    Matching coefficients confirms the values of the constants.

  11. Express the assumed form over a common denominator

    x+Ax−1+Bx+1=A(x+1)+B(x−1)x2−1x+\frac{A}{x - 1}+\frac{B}{x + 1}=\frac{A \left(x + 1\right) + B \left(x - 1\right)}{x^{2} - 1}

    Recombining shows the numerators must match.

  12. Verify by recombining the partial fractions

    x+1x−1=x3+1(x−1)(x+1)x + \frac{1}{x - 1}=\frac{x^{3} + 1}{\left(x - 1\right) \left(x + 1\right)}

    Adding the fractions back returns the original expression.

  13. Check the result at a test value

    x=0: −1=−1x=0:\ -1=-1

    Both sides agree, confirming the decomposition.

  14. Note the degrees involved

    deg⁡N=3, deg⁡D=2\deg N=3,\ \deg D=2

    The degree comparison determines whether division is needed.

  15. Select the correct decomposition

    x3+1(x−1)(x+1)=x+1x−1\frac{x^{3} + 1}{\left(x - 1\right) \left(x + 1\right)}=x + \frac{1}{x - 1}

    This is the fully correct partial fraction decomposition.

Answer
x+1x−1x + \frac{1}{x - 1}
Question 3
8 markschallenging
Which of the following is the correct partial fraction decomposition of x2+x+1x3−6x2+11x−6\frac{x^{2} + x + 1}{x^{3} - 6 x^{2} + 11 x - 6}?
Show worked solution

Worked solution

  1. Write the fraction to be decomposed

    x2+x+1x3−6x2+11x−6\frac{x^{2} + x + 1}{x^{3} - 6 x^{2} + 11 x - 6}

    We start from the given rational function.

  2. Factorise the denominator

    x3−6x2+11x−6=(x−3)(x−2)(x−1)x^{3} - 6 x^{2} + 11 x - 6=\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)

    Factorising shows which linear factors appear.

  3. Write the partial fraction form with unknown constants

    x2+x+1(x−3)(x−2)(x−1)=Ax−1+Bx−2+Cx−3\frac{x^{2} + x + 1}{\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}=\frac{A}{x - 1}+\frac{B}{x - 2}+\frac{C}{x - 3}

    Each distinct factor (and each power of a repeated factor) needs its own term.

  4. Cover up x - 1 and substitute x=1

    3=A (2) ⇒ A=323=A\,(2)\ \Rightarrow\ A=\frac{3}{2}

    Multiplying by the factor and substituting its root isolates one constant.

  5. Cover up x - 2 and substitute x=2

    7=B (−1) ⇒ B=−77=B\,(-1)\ \Rightarrow\ B=-7

    Multiplying by the factor and substituting its root isolates one constant.

  6. Cover up x - 3 and substitute x=3

    13=C (2) ⇒ C=13213=C\,(2)\ \Rightarrow\ C=\frac{13}{2}

    Multiplying by the factor and substituting its root isolates one constant.

  7. Multiply through by the denominator to form an identity

    x2+x+1=A(x2−5x+6)+B(x2−4x+3)+C(x2−3x+2)x^{2} + x + 1=A \left(x^{2} - 5 x + 6\right) + B \left(x^{2} - 4 x + 3\right) + C \left(x^{2} - 3 x + 2\right)

    Clearing the fractions gives an identity true for all x.

  8. Expand the right-hand side

    x2+x+1=Ax2−5Ax+6A+Bx2−4Bx+3B+Cx2−3Cx+2Cx^{2} + x + 1=A x^{2} - 5 A x + 6 A + B x^{2} - 4 B x + 3 B + C x^{2} - 3 C x + 2 C

    Expanding lets us compare like terms.

  9. Compare coefficients of like powers of x

    A+B+C=1, −5A−4B−3C=1, 6A+3B+2C=1A + B + C=1,\ - 5 A - 4 B - 3 C=1,\ 6 A + 3 B + 2 C=1

    Matching coefficients confirms the values of the constants.

  10. Express the assumed form over a common denominator

    Ax−1+Bx−2+Cx−3=A(x2−5x+6)+B(x2−4x+3)+C(x2−3x+2)x3−6x2+11x−6\frac{A}{x - 1}+\frac{B}{x - 2}+\frac{C}{x - 3}=\frac{A \left(x^{2} - 5 x + 6\right) + B \left(x^{2} - 4 x + 3\right) + C \left(x^{2} - 3 x + 2\right)}{x^{3} - 6 x^{2} + 11 x - 6}

    Recombining shows the numerators must match.

  11. Verify by recombining the partial fractions

    32(x−1)−7x−2+132(x−3)=x2+x+1(x−3)(x−2)(x−1)\frac{3}{2 \left(x - 1\right)} - \frac{7}{x - 2} + \frac{13}{2 \left(x - 3\right)}=\frac{x^{2} + x + 1}{\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}

    Adding the fractions back returns the original expression.

  12. Check the result at a test value

    x=0: −16=−16x=0:\ - \frac{1}{6}=- \frac{1}{6}

    Both sides agree, confirming the decomposition.

  13. Note the degrees involved

    deg⁡N=2, deg⁡D=3\deg N=2,\ \deg D=3

    The degree comparison determines whether division is needed.

  14. Recall the cover-up method

    Multiply by a factor, then substitute its root.\text{Multiply by a factor, then substitute its root.}

    The cover-up method is the quickest route to each constant.

  15. Select the correct decomposition

    x2+x+1(x−3)(x−2)(x−1)=32(x−1)−7x−2+132(x−3)\frac{x^{2} + x + 1}{\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}=\frac{3}{2 \left(x - 1\right)} - \frac{7}{x - 2} + \frac{13}{2 \left(x - 3\right)}

    This is the fully correct partial fraction decomposition.

Answer
32(x−1)−7x−2+132(x−3)\frac{3}{2 \left(x - 1\right)} - \frac{7}{x - 2} + \frac{13}{2 \left(x - 3\right)}
Question 4
8 markschallenging
Which of the following is the correct form of the partial fraction decomposition of x2+1x3−4x2+5x−2\frac{x^{2} + 1}{x^{3} - 4 x^{2} + 5 x - 2}?
Show worked solution

Worked solution

  1. Write the fraction to be decomposed

    x2+1x3−4x2+5x−2\frac{x^{2} + 1}{x^{3} - 4 x^{2} + 5 x - 2}

    We start from the given rational function.

  2. Factorise the denominator

    x3−4x2+5x−2=(x−2)(x−1)2x^{3} - 4 x^{2} + 5 x - 2=\left(x - 2\right) \left(x - 1\right)^{2}

    Factorising shows which linear factors appear.

  3. Write the partial fraction form with unknown constants

    x2+1(x−2)(x−1)2=Ax−1+B(x−1)2+Cx−2\frac{x^{2} + 1}{\left(x - 2\right) \left(x - 1\right)^{2}}=\frac{A}{x - 1}+\frac{B}{\left(x - 1\right)^{2}}+\frac{C}{x - 2}

    Each distinct factor (and each power of a repeated factor) needs its own term.

  4. Cover up \left(x - 1\right)^{2} and substitute x=1

    2=B (−1) ⇒ B=−22=B\,(-1)\ \Rightarrow\ B=-2

    Multiplying by the factor and substituting its root isolates one constant.

  5. Cover up x - 2 and substitute x=2

    5=C (1) ⇒ C=55=C\,(1)\ \Rightarrow\ C=5

    Multiplying by the factor and substituting its root isolates one constant.

  6. Substitute x=0 to find A

    x=0: A=−4x=0:\ A=-4

    With the other constants known, any convenient value gives the last one.

  7. Multiply through by the denominator to form an identity

    x2+1=A(x2−3x+2)+B(x−2)+C(x2−2x+1)x^{2} + 1=A \left(x^{2} - 3 x + 2\right) + B \left(x - 2\right) + C \left(x^{2} - 2 x + 1\right)

    Clearing the fractions gives an identity true for all x.

  8. Expand the right-hand side

    x2+1=Ax2−3Ax+2A+Bx−2B+Cx2−2Cx+Cx^{2} + 1=A x^{2} - 3 A x + 2 A + B x - 2 B + C x^{2} - 2 C x + C

    Expanding lets us compare like terms.

  9. Compare coefficients of like powers of x

    A+C=1, −3A+B−2C=0, 2A−2B+C=1A + C=1,\ - 3 A + B - 2 C=0,\ 2 A - 2 B + C=1

    Matching coefficients confirms the values of the constants.

  10. Express the assumed form over a common denominator

    Ax−1+B(x−1)2+Cx−2=A(x2−3x+2)+B(x−2)+C(x2−2x+1)x3−4x2+5x−2\frac{A}{x - 1}+\frac{B}{\left(x - 1\right)^{2}}+\frac{C}{x - 2}=\frac{A \left(x^{2} - 3 x + 2\right) + B \left(x - 2\right) + C \left(x^{2} - 2 x + 1\right)}{x^{3} - 4 x^{2} + 5 x - 2}

    Recombining shows the numerators must match.

  11. Verify by recombining the partial fractions

    −4x−1−2(x−1)2+5x−2=x2+1(x−2)(x−1)2- \frac{4}{x - 1} - \frac{2}{\left(x - 1\right)^{2}} + \frac{5}{x - 2}=\frac{x^{2} + 1}{\left(x - 2\right) \left(x - 1\right)^{2}}

    Adding the fractions back returns the original expression.

  12. Check the result at a test value

    x=0: −12=−12x=0:\ - \frac{1}{2}=- \frac{1}{2}

    Both sides agree, confirming the decomposition.

  13. Note the degrees involved

    deg⁡N=2, deg⁡D=3\deg N=2,\ \deg D=3

    The degree comparison determines whether division is needed.

  14. Recall the cover-up method

    Multiply by a factor, then substitute its root.\text{Multiply by a factor, then substitute its root.}

    The cover-up method is the quickest route to each constant.

  15. State the correct form of the decomposition

    Ax−1+B(x−1)2+Cx−2\frac{A}{x - 1}+\frac{B}{\left(x - 1\right)^{2}}+\frac{C}{x - 2}

    This form matches every factor of the denominator.

Answer
Ax−1+B(x−1)2+Cx−2\frac{A}{x - 1}+\frac{B}{\left(x - 1\right)^{2}}+\frac{C}{x - 2}
Question 5
8 markschallenging
Which of the following is the correct form of the partial fraction decomposition of x2+3x+1x2+x−2\frac{x^{2} + 3 x + 1}{x^{2} + x - 2}?
Show worked solution

Worked solution

  1. Write the fraction to be decomposed

    x2+3x+1x2+x−2\frac{x^{2} + 3 x + 1}{x^{2} + x - 2}

    We start from the given rational function.

  2. Compare the degrees of numerator and denominator

    deg⁡N=2 ≥ deg⁡D=2\deg N=2\ \ge\ \deg D=2

    The fraction is improper, so we must divide first.

  3. Divide the numerator by the denominator

    x2+3x+1x2+x−2=1+2x+3x2+x−2\frac{x^{2} + 3 x + 1}{x^{2} + x - 2}=1+\frac{2 x + 3}{x^{2} + x - 2}

    Polynomial division gives a polynomial part plus a proper fraction.

  4. Factorise the denominator

    x2+x−2=(x−1)(x+2)x^{2} + x - 2=\left(x - 1\right) \left(x + 2\right)

    Factorising shows which linear factors appear.

  5. Write the partial fraction form with unknown constants

    x2+3x+1(x−1)(x+2)=1+Ax−1+Bx+2\frac{x^{2} + 3 x + 1}{\left(x - 1\right) \left(x + 2\right)}=1+\frac{A}{x - 1}+\frac{B}{x + 2}

    Each distinct factor (and each power of a repeated factor) needs its own term.

  6. Cover up x - 1 and substitute x=1

    5=A (3) ⇒ A=535=A\,(3)\ \Rightarrow\ A=\frac{5}{3}

    Multiplying by the factor and substituting its root isolates one constant.

  7. Cover up x + 2 and substitute x=-2

    −1=B (−3) ⇒ B=13-1=B\,(-3)\ \Rightarrow\ B=\frac{1}{3}

    Multiplying by the factor and substituting its root isolates one constant.

  8. Multiply through by the denominator to form an identity

    2x+3=A(x+2)+B(x−1)2 x + 3=A \left(x + 2\right) + B \left(x - 1\right)

    Clearing the fractions gives an identity true for all x.

  9. Expand the right-hand side

    2x+3=Ax+2A+Bx−B2 x + 3=A x + 2 A + B x - B

    Expanding lets us compare like terms.

  10. Compare coefficients of like powers of x

    A+B=2, 2A−B=3A + B=2,\ 2 A - B=3

    Matching coefficients confirms the values of the constants.

  11. Express the assumed form over a common denominator

    1+Ax−1+Bx+2=A(x+2)+B(x−1)x2+x−21+\frac{A}{x - 1}+\frac{B}{x + 2}=\frac{A \left(x + 2\right) + B \left(x - 1\right)}{x^{2} + x - 2}

    Recombining shows the numerators must match.

  12. Verify by recombining the partial fractions

    1+13(x+2)+53(x−1)=x2+3x+1(x−1)(x+2)1 + \frac{1}{3 \left(x + 2\right)} + \frac{5}{3 \left(x - 1\right)}=\frac{x^{2} + 3 x + 1}{\left(x - 1\right) \left(x + 2\right)}

    Adding the fractions back returns the original expression.

  13. Check the result at a test value

    x=0: −12=−12x=0:\ - \frac{1}{2}=- \frac{1}{2}

    Both sides agree, confirming the decomposition.

  14. Note the degrees involved

    deg⁡N=2, deg⁡D=2\deg N=2,\ \deg D=2

    The degree comparison determines whether division is needed.

  15. State the correct form of the decomposition

    1+Ax−1+Bx+21+\frac{A}{x - 1}+\frac{B}{x + 2}

    This form matches every factor of the denominator.

Answer
1+Ax−1+Bx+21+\frac{A}{x - 1}+\frac{B}{x + 2}

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