Practise dividing cubic or quartic polynomials by linear or quadratic divisors and identifying quotient and remainder through long division, inspection or coefficient comparison.
Find the quotient and the remainder when 3x4−2x2 is divided by x+1.
Commence division and reach partial quotient of the form 3 x^3 +/- 3 x^2
or 3 x^4-2 x^2 (x+1)(A x^3+B x^2+C x+D)+E x+F, and reach A=3 and B= +/- 3
M1
May be seen in synthetic division.
Obtain quotient 3 x^3-3 x^2+x-1 Do not ISW
A1
Don't need to state which is the quotient and which
is remainder. However, if clearly muddled, then
M1A1A0 for both expressions correct.
Obtain remainder of 1
A1
Do not ISW.
Marking guidance:
Allow e.g. 3 x^3-3 x^2+x-1+1/x+1 but NOT
remainder =1/x+1.
Alternative Method for Question 2
f(-1)=3-2=1= remainder
B1
Do not ISW.
Use division or inspection or compare coefficients
M1
3 x^4-2 x^2-1 (x+1)(3 x^3-3 x^2+x-1)
Obtain quotient 3 x^3-3 x^2+x-1
A1
Do not ISW.
Allow e.g. 3 x^3-3 x^2+x-1+1/x+1 but NOT
remainder =1/x+1.