2021HSCShort answer

Q11 (f)

3 marks

Question

Express $\frac{3x^2-5}{(x-2)(x^2+x+1)}$ as a sum of partial fractions over $\mathbb R$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Evaluates one of the three coefficients	2
Provides a correct expression for the sum of fractions with unknown coefficients, or equivalent merit	1

Sample answer:

$\frac{3x^2-5}{(x-2)(x^2+x+1)}=\frac A{x-2}+\frac{Bx+C}{x^2+x+1}$
$3x^2-5=A(x^2+x+1)+(Bx+C)(x-2)$

When $x=2$ :

$3(2)^2-5=A(2^2+2+1)$
$7=7A,\qquad A=1$

Equating coefficients of $x^2$ :

$3=A+B\quad\therefore B=2$

Equating constants:

$-5=A-2C$
$2C=6,\qquad C=3$

$\therefore\frac{3x^2-5}{(x-2)(x^2+x+1)}=\frac1{x-2}+\frac{2x+3}{x^2+x+1}$

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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