Paper 2 · June 2017 · Algebraic manipulation

Express $\dfrac{3x^{2}+6x+7}{x^{2}+2x+1}$ in the form $\dfrac{\mathrm{A}}{(x+\mathrm{B})^{2}}+\mathrm{C}$ , stating the values of the constants A, B and C.

Model answer

$\mathrm{A}=4$ , $\mathrm{B}=1$ , $\mathrm{C}=3$

Also accepted: A=4, B=1, C=3, 4, 1, 3, A=4 B=1 C=3

Explanation

The denominator factorises as

x^{2}+2x+1=(x+1)^{2}

, and the numerator can be written

3(x^{2}+2x+1)+4=3(x+1)^{2}+4

. Dividing,

\dfrac{3(x+1)^{2}+4}{(x+1)^{2}}=\dfrac{4}{(x+1)^{2}}+3

, so

\mathrm{A}=4

\mathrm{B}=1

and

\mathrm{C}=3

Derived from ZIMSEC Mathematics Paper 2, June 2017, Q1

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