Find the values of p and q such that (x – 1)and (x – 3) are factors of px3 + qx2 + 11x – 6

A.
1, 6 
B.
1, 6 
C.
1, 6 
D.
6, 1
Correct Answer: Option B
Explanation
Since (x – 1), is a factor, when the polynomial is divided by (x – 1), the remainder = zero
(therefore (x – 1) = 0)
x = 1
Substitute in the polynomial the value x = 1
= (p(1)^3 + q(1)^2 + 11(1) – 6 = 0)
p + q + 5 = 0 …..(i)
Also since x – 3 is a factor, (therefore) x – 3 = 0
x = 3
Substitute (p(3)^3 + q(3)^2 + 11(3) – 6 = 0)
27p + 9q = 27 ……(2)
Combine eqns. (i) and (ii)
Multiply equation (i) by 9 to eliminate q
9p + 9q = 45
Subtract (ii) from (i), (18p = 18)
(therefore) p = 1
Put p = 1 in (i),
(1 + q = 5 implies q = 6)
((p, q) = (1, 6))