Single unknown from factor condition

Questions where one unknown constant is found using a single factor condition, then factorise and prove root count by examining the discriminant of the quadratic factor.

6 questions · Moderate -0.6

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

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CAIE P3 2013 November Q3

5 marks Moderate -0.8

3 The polynomial $\mathrm { f } ( x )$ is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where $a$ is a constant. It is given that ( $x + 2$ ) is a factor of $\mathrm { f } ( x )$.

Find the value of $a$.
Show that, when $a$ has this value, the equation $\mathrm { f } ( x ) = 0$ has only one real root.

Edexcel C12 2019 January Q13

10 marks Moderate -0.3

13. $\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12$, where $c$ is a constant Given that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$,

show that $c = - 14$
Write $\mathrm { f } ( x )$ in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where $\mathrm { Q } ( x )$ is a quadratic function.
Use the answer to part (b) to prove that the equation $\mathrm { f } ( x ) = 0$ has only one real solution. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. On separate diagrams sketch the curve with equation
1. $y = \mathrm { f } ( 3 x )$
2. $y = - \mathrm { f } ( \mathrm { x } )$ On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.

Edexcel P2 2023 June Q2

6 marks Moderate -0.3

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } + 5 x + a$$ where $a$ is a constant.
Given that ( $2 x - 3$ ) is a factor of $\mathrm { f } ( x )$,

use the factor theorem to show that $a = - 3$
Hence show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.

Pre-U Pre-U 9794/1 Specimen Q3

5 marks Moderate -0.3

Find the value of $a$ for which ( $x - 2$ ) is a factor of $5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8$.
Show that, for this value of $a$, the cubic equation $5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8 = 0$ has only one real root.

Edexcel C2 Q13

7 marks Moderate -0.8

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that $f(4) = 0$,

Find the value of $c$, [2]
factorise $f(x)$ as the product of a linear factor and a quadratic factor. [3]
Hence show that, apart from $x = 4$, there are no real values of $x$ for which $f(x) = 0$. [2]

Edexcel C2 Q4

7 marks Moderate -0.8

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that $f(4) = 0$,

find the value of $c$, [2]
factorise $f(x)$ as the product of a linear factor and a quadratic factor. [3]
Hence show that, apart from $x = 4$, there are no real values of $x$ for which $f(x) = 0$. [2]