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Chloe is attempting to write \( \frac{2x^2 + x}{(x + 1)(x + 2)^2} \) as partial fractions, with constant numerators - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 1
Question 9
Chloe is attempting to write \( \frac{2x^2 + x}{(x + 1)(x + 2)^2} \) as partial fractions, with constant numerators. Her incorrect attempt is shown below. Step 1 \... show full transcript
Worked Solution & Example Answer:Chloe is attempting to write \( \frac{2x^2 + x}{(x + 1)(x + 2)^2} \) as partial fractions, with constant numerators - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 1
Step 1
9 (a) (i) By using a counter example, show that the answer obtained by Chloe cannot be correct.
Answer
To demonstrate that Chloe's answer is incorrect, we can substitute a specific value for ( x ). Let's use ( x = -2 ):
Calculating the left-hand side (LHS): [ LHS = \frac{2(-2)^2 + (-2)}{((-2) + 1)((-2) + 2)^2} = \frac{8 - 2}{(-1)(0)} ] This expression implies a division by zero, which is undefined.
Now, considering the right-hand side (RHS): [ RHS = \frac{1}{-2 + 1} + \frac{6}{(-2 + 2)^2} = \frac{1}{-1} + \frac{6}{0} ] Both components of the RHS also lead to undefined results.
Since both sides are undefined for ( x = -2 ), it confirms that Chloe's expression cannot be correct overall.
Step 2
9 (a) (ii) Explain her mistake in Step 1.
Answer
Chloe should have included an additional term with ( x + 2 ) in the denominator. Specifically, she neglected to account for the fact that partial fractions with repeated factors require an additional constant term for each power of the factor.
Step 3
9 (b) Write \( \frac{2x^2 + x}{(x + 1)(x + 2)^2} \) as partial fractions, with constant numerators.
Answer
To express ( \frac{2x^2 + x}{(x + 1)(x + 2)^2} ) in partial fractions, we will assume:
[ \frac{2x^2 + x}{(x + 1)(x + 2)^2} = \frac{A}{(x + 1)} + \frac{B}{(x + 2)} + \frac{C}{(x + 2)^2} ]
Combining the right-hand side gives:
[ 2x^2 + x = A(x + 2)^2 + B(x + 1)(x + 2) + C(x + 1) ]
Expanding and equating coefficients, we can find values for A, B, and C.
After resolving, we find:
( A = 1, B = 6, C = -4 ).
Thus, the expression becomes:
[ \frac{2x^2 + x}{(x + 1)(x + 2)^2} = \frac{1}{(x + 1)} + \frac{6}{(x + 2)} - \frac{4}{(x + 2)^2} ].