CAIE A-Level Mathematics 3.6.1 Locating Roots

CAIE A-Level Mathematics 3.6.1 Locating Roots
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise locating roots with graph intersections or a sign change, calculating both endpoints and explaining why the stated interval contains a root or exactly one root.

How this is tested

  • evaluate the same continuous function at both endpoints and state the opposite signs
  • conclude that a root lies inside the interval rather than reporting endpoint values alone
  • sketch both sides with correct shape and placement to justify a unique intersection

Question 7(c)

[Maximum number: 2]

The polynomial p(x) is defined by

p(x)=2x4+kx3+kx2+17x+18,\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

Show by calculation that −1.4<β<−1.0-1.4<\beta<-1.0.