The polynomial \({x^4} + p{x^3} + q{x^2} + rx + 6\) is exactly divisible by each of \(\left( {x - 1} \right)\), \(\left( {x - 2} \right)\) and \(\left( {x - 3} \right)\).

Find the values of \(p\), \(q\) and \(r\).

METHOD 1

substitute each of \(x\) = 1,2 and 3 into the quartic and equate to zero      (M1)

\(p + q + r =  - 7\)

\(4p + 2q + r =  - 11\) or equivalent        (A2)

\(9p + 3q + r =  - 29\)

Note: Award A2 for all three equations correct, A1 for two correct.

attempting to solve the system of equations      (M1)

\(p\) = −7, \(q\) = 17, \(r\) = −17     A1

Note: Only award M1 when some numerical values are found when solving algebraically or using GDC.

METHOD 2

attempt to find fourth factor      (M1)

\(\left( {x - 1} \right)\)     A1

attempt to expand \({\left( {x - 1} \right)^2}\left( {x - 2} \right)\left( {x - 3} \right)\)     M1

\({x^4} - 7{x^3} + 17{x^2} - 17x + 6\) (\(p\) = −7, \(q\) = 17, \(r\) = −17)     A2

Note: Award A2 for all three values correct, A1 for two correct.

Note: Accept long / synthetic division.

[5 marks]