Prove root count with given polynomial - Factor & Remainder Theorem

6 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$ where $a$ is a constant.

Given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$, find the value of $a$.
When $a$ has this value, find the quotient when $\mathrm { p } ( x )$ is divided by ( $x + 2$ ) and hence show that the equation $\mathrm { p } ( x ) = 0$ has exactly one real root.

4. $$f ( x ) = \left( x ^ { 2 } - 2 \right) ( 2 x - 3 ) - 21$$

State the value of the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x - 3$ )
Use the factor theorem to show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$
Hence,
1. factorise $\mathrm { f } ( x )$
2. show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.

5 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6$.

Find the remainder when $\mathrm { f } ( x )$ is divided by $( x - 3 )$.
Given that $\mathrm { f } ( 2 ) = 0$, express $\mathrm { f } ( x )$ as the product of a linear factor and a quadratic factor.
Determine the number of real roots of the equation $\mathrm { f } ( x ) = 0$, giving a reason for your answer.

11. $$f ( x ) = 2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 48$$

Prove that $( x - 4 )$ is a factor of $\mathrm { f } ( x )$.
Hence, using algebra, show that the equation $\mathrm { f } ( x ) = 0$ has only two distinct roots. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-24_727_1059_566_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$.
Deduce, giving reasons for your answer, the number of real roots of the equation $$2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 46 = 0$$ Given that $k$ is a constant and the curve with equation $y = \mathrm { f } ( x + k )$ passes through the origin, (d) find the two possible values of $k$.

2. $$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$

Use the factor theorem to show that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$.
Find the constants $a$, $b$ and $c$ such that $$f ( x ) = ( x + 3 ) \left( a x ^ { 2 } + b x + c \right)$$
Hence show that $\mathrm { f } ( x ) = 0$ has only one real root.
Write down the real root of the equation $\mathrm { f } ( x - 5 ) = 0$

4. $$f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } + 2 x - 6$$

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

6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$

1. Calculate f(2)
2. Write $\mathrm { f } ( x )$ as a product of two algebraic factors. Using the answer to (a)(ii),
prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
deduce the number of real solutions, for $7 \pi \leqslant \theta < 10 \pi$, to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$

6 The polynomial $x ^ { 3 } - 4 x ^ { 2 } + 10 x - 21$ is denoted by $\mathrm { f } ( x )$.

Use the factor theorem to show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$.
The polynomial $\mathrm { f } ( x )$ can be written as $( \mathrm { x } - 3 ) \left( \mathrm { x } ^ { 2 } + \mathrm { bx } + \mathrm { c } \right)$ where $b$ and $c$ are constants. Find the values of $b$ and $c$.
Show that $x = 3$ is the only real root of the equation $\mathrm { f } ( x ) = 0$.

6

The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + x - 10$.
1. Use the Factor Theorem to show that $x - 2$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x )$ in the form $( x - 2 ) \left( x ^ { 2 } + a x + b \right)$, where $a$ and $b$ are constants.
The curve $C$ with equation $y = x ^ { 3 } + x - 10$, sketched below, crosses the $x$-axis at the point $Q ( 2,0 )$.
\includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
1. Find the gradient of the curve $C$ at the point $Q$.
2. Hence find an equation of the tangent to the curve $C$ at the point $Q$.
3. Find $\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x$.
4. Hence find the area of the shaded region bounded by the curve $C$ and the coordinate axes.

6 The cubic polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = ( x - 2 ) \left( x ^ { 2 } + x + 3 \right)$.

Show that $\mathrm { p } ( x )$ can be written in the form $x ^ { 3 } + a x ^ { 2 } + b x - 6$, where $a$ and $b$ are constants whose values are to be found.
Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
(2 marks)
Prove that the equation $( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0$ has only one real root and state its value.
(3 marks)

1. Answer all the questions. $$f ( x ) = 3 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 10$$

Use the factor theorem to show that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$
Find the values of the constants $a , b$ and $c$ such that $$\mathrm { f } ( x ) \equiv ( x - 2 ) \left( a x ^ { 2 } + b x + c \right)$$
Using your answer to part (b) show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.
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6

The polynomial $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5$.
1. Use the Factor Theorem to show that $x - 1$ is a factor of $\mathrm { f } ( x )$.
2. Express $\mathrm { f } ( x )$ in the form $( x - 1 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
3. Hence show that the equation $\mathrm { f } ( x ) = 0$ has exactly one real root and state its value.
The curve with equation $y = x ^ { 3 } + 4 x - 5$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the $x$-axis at the point $A ( 1,0 )$ and the point $B ( 2,11 )$ lies on the curve.
1. Find $\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve and the line $A B$.

4

Use the factor theorem to prove that $x - 6$ is a factor of $\mathrm { p } ( x )$.
$4 \quad \mathrm { p } ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } - 48 x - 36$ 4
1. Prove that the graph of $y = \mathrm { p } ( x )$ intersects the $x$-axis at exactly one point.
  4
(ii) State the coordinates of this point of intersection.