Standard +0.8 This is a Further Maths question requiring knowledge that complex roots come in conjugate pairs for polynomials with real coefficients, then forming a quadratic factor and performing polynomial division to find remaining roots. While systematic, it involves multiple algebraic steps (forming conjugate pair, multiplying to get quadratic, long division, solving resulting quadratic) making it moderately challenging but still a standard FM technique.
3.
$$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$
Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\)
(Solutions based entirely on graphical or numerical methods are not acceptable.)
3.
$$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$
Given that $x = - 1 + 3 \mathrm { i }$ is a root of the equation $\mathrm { f } ( x ) = 0$, use algebra to find the three other roots of $\mathrm { f } ( x ) = 0$\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)\\
\hfill \mbox{\textit{Edexcel F1 2017 Q3 [7]}}