f(x)=2x3+(3−2k)x2−(3k+2)x+2kf(x)=2x^3+(3-2k)x^2-(3k+2)x+2kf(x)=2x3+(3−2k)x2−(3k+2)x+2k, where k k\,k is a constant and k>12k>\dfrac12k>21.
Use the factor theorem to show that (x+2)(x + 2)(x+2) is a factor of f(x)f(x)f(x)
Hence, showing all your working, write f(x)f(x)f(x) as a product of three linear factors in terms of kkk.
The finite region R R\,R is bounded by the curve with equation y=f(x)y = f(x)y=f(x) and the xxx-axis, and lies below the xxx-axis. Find, using algebraic integration, the exact value of the area of R R\,R in terms of kkk.
411 exam-style questions on Edexcel A Level Maths Integration, covering 13.1 Integrating x^n, 13.2 Indefinite Integrals, 13.3 Finding Functions, 13.4 Definite Integrals, 13.5 Areas under Curves, 13.6 Areas under the x-axis, and 13.7 Areas between curves and lines. Each one has a worked solution and a mark scheme showing where the marks go.