The polynomial p (x) is defined by

             p(x) = 4x3 + 16x2 + 9x -15.

(a)Find the quotient when p(x) is divided by (2x+3), and show that the remainder is −6.

(b) Find \(\int \frac{p(x)}{2x+3}dx\) .

(c)Factorise p (x) +6 completely and hence solve the equation

                    p (cosec 2θ) + 6 = 0

for 00 < θ < 1350.



(a)Carry out division at least as far as 2x2 + kx

Obtain quotient 2x2 + 5x – 3

Confirm remainder is -6.

(b)Integrate to obtain at least k1 x 3 and  k2 ln (2x 3) + terms

Obtain \(\frac{2}{3}x^{3} + \frac{5}{2}x^{2} – 3x – 3 In (2x+3)\)

(c)State or imply p(x) + 6 = (2x + 3) (2x2 + 5x -3)

Conclude (2x + 3) (2x – 1) (x + 3)

State or imply  sin 2θ = \(-\frac{2}{3}\) or sin 2θ = \(-\frac{1}{3}\) or both

Carry out correct process to find θ in at least one case

Obtain 99.7 and 110.9


Given that \(3e^{x}+8e^{-x}=14\) find the possible values of \(e^{x}\) and hence solve the equation \(3e^{x}+8e^{-x}=14\) correct to 3 significant figures.


Rearrange to \(3e^{2x}-14e^{x}+8=0\) or equivalent involving substitution Solve quadratic equation in\( e^{x}\) to find two values of \( e^{x}\) Obtain\( \frac{2}{3} and 4\)

Use natural logarithms to solve equation of form\( e^{x}\) =k where k > 0 dep on Allow M mark if left in exact form

Obtain −0.405

Obtain 1.39


Show that ln \(\left ( x^{3} -4x\right )In\left ( x+2 \right )\)Use logarithm subtraction property to produce logarithm of quotient M1


Factorise at least as far as  \(x\left ( x^{2} -4\right )\) and  \(x\left ( x-2 \right ) \) or use correct algebraic long division to obtain a quotient of x+2 and a remainder5 of 0 from correct working .

Obtain final answer \(\ln \left ( x+2 \right )\) using correct process.