CAIE A-Level Mathematics 3.1.3 Factor and Remainder Theorems

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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.

How this is tested

substitute the root of ax + b into p(x) and equate the result to zero or the remainder
use p(a) = 0 together with p′(a) = 0 when a repeated linear factor is given
divide and factorise completely, then use a sign chart to solve any polynomial inequality

[Maximum number: 6]

The polynomial p(x) is defined by

\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).

(a)

Find the value of k.

It is given that the equation p(x)=0 has exactly two real roots, denoted by $\alpha$ and $\beta$ , where $\alpha$ is an integer and $\beta$ is not an integer.

[ 2 ]

(b)

State the value of $\alpha$ and show that $\beta$ satisfies the equation $x=\sqrt[3]{-2 x-4.5}$ .

[ 4 ]

How this is tested

Question 7

Question 7(a)

Question 7(b)