Practise applying p(a) = 0 or p(a) = r, using derivative conditions for repeated factors and dividing known factors to solve coefficients, equations and polynomial inequalities.
The polynomial 3x3+pax2+7a2x+qa3 is denoted by f(x), where p, q and a are constants and a=0.
When f(x) is divided by (x+2 a) the remainder is −22a3. When f(x) is divided by (3 x-a) the remainder is −a3.
Find the values of p and q.
Use f(-2 a)=-22 a^3
M1
Or use long division and equate a constant
remainder to -22 a^3.
Obtain -24 a^3+4 p a^3-14 a^3+q a^3=-22 a^3
A1
Must evaluate the terms.
OE, e.g. 4 p+q=16.
Use f(a/3)=-a^3
M1
Or use long division and equate a constant
remainder to -a^3.
Obtain 1/9 a^3+1/9 p a^3+7/3 a^3+q a^3=-a^3
A1
Must evaluate the terms.
OE, e.g. p+9 q=-31
Obtain p=5, q=-4
A1