Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The polynomial $p(x)$ is specified by $p(x) = 2x^4 + kx^3 + kx^2 + 17x + 18$, with $k$ a constant. You are told that $(x + 2)$ is a factor of $p(x)$. You are also told that the equation $p(x) = 0$ has precisely two real roots, named $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer.
(a)[2]
Determine the value of $k$.
(b)[4]
State the value of $\alpha$ and demonstrate that $\beta$ satisfies the equation $x = \sqrt[3]{-2x - 4.5}$.
(c)[2]
Show, using calculation, that $-1.4<\beta<-1.0$.
(d)[3]
Use an iterative formula, based on the equation in part (b), to determine the value of $\beta$ correct to 3 significant figures. State each iteration to 5 significant figures.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use $x=-2$, set the expression equal to zero and make an attempt at the solution” …