CAIE A-Level Mathematics 2.1.3 Factor and Remainder Theorems

CAIE A-Level Mathematics 2.1.3 Factor and Remainder Theorems
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise evaluating p(a) for factors and remainders, forming simultaneous equations for unknown coefficients and dividing by known linear factors to complete polynomial solutions.

How this is tested

  • substitute the zero of ax + b into p(x), equating the result to zero for a factor
  • translate each stated remainder into p(a) = r and solve the resulting coefficient equations
  • divide by a known factor and factorise the quotient, then classify any remaining real roots

Question 5(a)

[Maximum number: 4]

The polynomial p(x) is defined by

p(x)=ax4+bx3+13x2−35x+15,\mathrm{p}(x)=a x^{4}+b x^{3}+13 x^{2}-35 x+15,

where a and b are constants. It is given that ( 2 x-1 ) and ( x-3 ) are factors of p(x).

Find the values of a and b.