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]