Write down the quadratic expression \(2{x^2} + x - 3\) as the product of two linear factors.

Hence, or otherwise, find the coefficient of \(x\) in the expansion of \({\left( {2{x^2} + x - 3} \right)^8}\) .

\(2{x^2} + x - 3 = \left( {2x + 3} \right)\left( {x - 1} \right)\)     A1

Note: Accept \(2\left( {x + \frac{3}{2}} \right)\left( {x - 1} \right)\).

Note: Either of these may be seen in (b) and if so A1 should be awarded.

[1 mark]

EITHER

\({\left( {2{x^2} + x - 3} \right)^8} = {\left( {2x + 3} \right)^8}{\left( {x - 1} \right)^8}\)     M1

\( = \left( {{3^8} + 8\left( {{3^7}} \right)\left( {2x} \right) + ...} \right)\left( {{{\left( { - 1} \right)}^8} + 8{{\left( { - 1} \right)}^7}\left( x \right) + ...} \right)\)     (A1)

coefficient of \(x = {3^8} \times 8 \times {\left( { - 1} \right)^7} + {3^7} \times 8 \times 2 \times {\left( { - 1} \right)^8}\)     M1

= −17 496     A1

Note: Under ft, final A1 can only be achieved for an integer answer.

OR

\({\left( {2{x^2} + x - 3} \right)^8} = {\left( {3 - \left( {x - 2{x^2}} \right)} \right)^8}\)     M1

\( = {3^8} + 8\left( { - \left( {x - 2{x^2}} \right)\left( {{3^7}} \right) + ...} \right)\)     (A1)

coefficient of \(x = 8 \times \left( { - 1} \right) \times {3^7}\)     M1

= −17 496     A1

Note: Under ft, final A1 can only be achieved for an integer answer.

[4 marks]

Many candidates struggled to find an efficient approach to this problem by applying the Binomial Theorem. A disappointing number of candidates attempted the whole expansion which was clearly an unrealistic approach when it is noted that the expansion is to the 8^th power. The fact that some candidates wrote down Pascal’s Triangle suggested that they had not studied the Binomial Theorem in enough depth or in a sufficient variety of contexts.

Many candidates struggled to find an efficient approach to this problem by applying the Binomial Theorem. A disappointing number of candidates attempted the whole expansion which was clearly an unrealistic approach when it is noted that the expansion is to the 8^th power. The fact that some candidates wrote down Pascal’s Triangle suggested that they had not studied the Binomial Theorem in enough depth or in a sufficient variety of contexts.