Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The polynomial $p(x)$ is given 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 exactly two real roots, called $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer.
(a)[2]
Calculate the value of $k$.
(b)[4]
State the value of $\alpha$ and demonstrate that $\beta$ satisfies $x = \sqrt[3]{-2x - 4.5}$.
(c)[2]
Show, by calculation, that $-1.4<\beta<-1.0$.
(d)[3]
Use an iterative formula, based on the equation in part (b), to determine $\beta$ correct to 3 significant figures. Record 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: “Substitute $x=-2$, set the result to zero and try to solve” …