For the polynomial equation \(p(x) = 0\), state

(i)     the sum of the roots;

(ii)     the product of the roots.

A new polynomial is defined by \(q(x) = p(x + 4)\).

Find the sum of the roots of the equation \(q(x) = 0\).

(i)     \(\left( { - \frac{{{a_{n - 1}}}}{{{a_n}}} = } \right) - \frac{1}{2}\)     A1

(ii)     \(\left( {{{( - 1)}^n}\frac{{{a_0}}}{{{a_n}}} = } \right) - \frac{{36}}{2} = ( - 18)\)     A1A1

Note:     First A1 is for the negative sign.

[3 marks]

METHOD 1

if \(\lambda \) satisfies \(p(\lambda ) = 0\) then \(q(\lambda  - 4) = 0\)

so the roots of \(q(x)\) are each \(4\) less than the roots of \(p(x)\)     (R1)

so sum of roots is \( - \frac{1}{2} - 4 \times 5 =  - 20.5\)     A1

METHOD 2

\(p(x + 4) = 2{x^5} + 2 \times 5 \times 4{x^4} \ldots  + {x^4} \ldots  = 2{x^5} + 41{x^4} \ldots \)     (M1)

so sum of roots is \( - \frac{{41}}{2} =  - 20.5\)     A1

[2 marks]

Tofal [5 marks]

Both parts fine if they used the formula, some tried to use the quadratic equivalent formula. Surprisingly some even found all the roots.

Some notation problems for weaker candidates. Good candidates used either of the methods shown in the Markscheme.