18M.2.AHL.TZ2.H_2

pestleMathematicsAAHLPaper 218M· ahl-2-12-factor-and-remainder-theorems-sum-and-product-of-rootssource ↗

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 $r$ .

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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]

Examiners’ report

[N/A]