Introduction
This partial fractions question is a classic O Level A Maths algebra skill that students must know well. Many students understand the final form, but they lose marks when they set up the wrong decomposition or compare coefficients incorrectly. Once the denominator is split properly, this partial fractions question becomes very structured and manageable.
The Question / Scenario Explanation
Source: Partial Fractions O Level 2024 Paper 1 A.Maths
Question (as shown): Express \(\frac{3x-5}{x^2(x-1)}\) in partial fractions.
Step-by-Step Solution / Explanation
Step 1: Write the correct partial fractions form
Since the denominator is \(x^2(x-1)\), we have:
\(\frac{3x-5}{x^2(x-1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-1}\)
This is the correct starting point for this partial fractions question because:
- \(x^2\) is a repeated linear factor, so we need both \(\frac{A}{x}\) and \(\frac{B}{x^2}\)
- \((x-1)\) is another linear factor, so we include \(\frac{C}{x-1}\)
Step 2: Clear the denominator
Multiply both sides by \(x^2(x-1)\):
\(3x-5=A x(x-1)+B(x-1)+C x^2\)
Step 3: Expand the right-hand side
Expand each term carefully:
\(A x(x-1)=Ax^2-Ax\)
\(B(x-1)=Bx-B\)
\(Cx^2=Cx^2\)
So:
\(3x-5=(A+C)x^2+(-A+B)x-B\)
Step 4: Compare coefficients
Now compare both sides:
Left side: \(3x-5\)
Right side: \((A+C)x^2+(-A+B)x-B\)
Matching coefficients gives:
\(A+C=0\)
\(-A+B=3\)
\(-B=-5\)
Step 5: Solve for the constants
From \(-B=-5\), we get:
\(B=5\)
Substitute into \(-A+B=3\):
\(-A+5=3\)
\(-A=-2\)
\(A=2\)
Then use \(A+C=0\):
\(2+C=0\)
\(C=-2\)
Step 6: Write the final answer
Substitute the values back into the decomposition:
\(\frac{3x-5}{x^2(x-1)}=\frac{2}{x}+\frac{5}{x^2}-\frac{2}{x-1}\)
✅ Final Answer: \(\frac{2}{x}+\frac{5}{x^2}-\frac{2}{x-1}\)
Key Concepts Students Must Know
- In a partial fractions question, repeated linear factors must be written with separate terms, such as \(\frac{A}{x}+\frac{B}{x^2}\).
- Always choose the decomposition form based on the denominator first before doing any algebra.
- After clearing the denominator, expand carefully and compare coefficients of like powers of \(x\).
- For O Level A Maths, students should be comfortable with both repeated factors and distinct linear factors in partial fractions.
Exam Tips / Common Mistakes
Exam Tips
- Write the decomposition form correctly before starting the working.
- For this partial fractions question, remember that \(x^2\) means you need both \(\frac{A}{x}\) and \(\frac{B}{x^2}\).
- Expand one term at a time to avoid sign mistakes.
- Line up coefficients clearly for \(x^2\), \(x\), and constants.
- Check your final constants by substituting them back into the expression.
Common Mistakes
- Writing \(\frac{A}{x^2}+\frac{B}{x-1}\) only, and forgetting the \(\frac{A}{x}\) term for the repeated factor.
- Expanding \(A x(x-1)\) incorrectly.
- Getting the sign wrong when comparing \(-B=-5\).
- Finding correct values for \(A\), \(B\), and \(C\) but copying the final partial fractions expression wrongly.
Parent Insight
This partial fractions question is a strong example of how O Level A Maths often combines method and accuracy. Students may know the topic, but still lose marks through small setup errors or careless expansion. With guided practice, children learn to recognise the denominator pattern quickly, set up the correct decomposition, and solve the constants with much more confidence.
Conclusion
To solve this partial fractions question, we first wrote \(\frac{3x-5}{x^2(x-1)}\) as \(\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-1}\). After clearing the denominator and comparing coefficients, we found \(A=2\), \(B=5\), and \(C=-2\). So the final answer is \(\frac{2}{x}+\frac{5}{x^2}-\frac{2}{x-1}\).
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