Standard +0.3 This is a straightforward Further Maths question requiring standard techniques: using the conjugate root theorem (since coefficients are real, 3+5i is also a root), forming a quadratic factor from the complex conjugate pair, then polynomial division to find the remaining quadratic factor. While it involves multiple steps and careful arithmetic, it follows a well-established algorithm with no novel insight required. Slightly above average difficulty due to the computational demands and being Further Maths content.
3 In this question you must show detailed reasoning.
The equation \(x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0\) has a root \(3 - 5 i\).
Find the other three roots of this equation.
\(3+5i\) may be mentioned as a root earlier in the solution
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3 + 5i$ is a root | B1 | Need to see statement that $3+5i$ is a root. May happen at end of question |
| Attempt to expand $(x-(3+5i))(x-(3-5i))$ | M1 | Attempt to use conjugate pair to derive a real quadratic. May see $(3+5i)(3-5i)=9+25=34$ and $(3+5i)+(3-5i)=6$ instead of expansion |
| $= x^2 - 6x + 34$, so this must be a factor | A1 | |
| $x^4 - 7x^3 - 2x^2 + 218x - 1428 = (x^2 - 6x + 34)(x^2 + \ldots x - 42)$ or $(x^2-6x+34)(x^2-x+\ldots)$ | M1 | Attempt to factorise or divide resulting in $x^2$ and one other term. **NB: This question required detailed reasoning** |
| $(x^2 - 6x + 34)(x^2 - x - 42)$ | A1 | |
| $(x^2 - x - 42) = (x-7)(x+6) \Rightarrow$ roots $-6, 7$ (and $3+5i$) | A1 | $3+5i$ may be mentioned as a root earlier in the solution |
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3 In this question you must show detailed reasoning.\\
The equation $x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0$ has a root $3 - 5 i$.\\
Find the other three roots of this equation.
\hfill \mbox{\textit{OCR Further Pure Core AS 2021 Q3 [6]}}