Prove root count with given polynomial

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

1. Given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. 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. $$f ( x ) = \left( x ^ { 2 } - 2 \right) ( 2 x - 3 ) - 21$$

1. State the value of the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) )
2. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\)
3. 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 \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).

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

7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

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

1. Prove that \(( x - 4 )\) is a factor of \(\mathrm { f } ( x )\).
2. 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 \(y = \mathrm { f } ( x )\).
3. 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. $$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$

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

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

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

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

6

1. 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.
2. 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.

5

1. 1. Sketch the curve with equation \(y = x ( x - 2 ) ^ { 2 }\).
   2. Show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   2. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x - 3 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
3. Hence show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) has only one real root and state the value of this root.

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

1. 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.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   (2 marks)
3. Prove that the equation \(( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0\) has only one real root and state its value.
   (3 marks)

5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
2. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
3. 1. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   2. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

7

1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]

6

1. 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.
2. 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 \(A B\).

\(p(x) = 4x^3 - 15x^2 - 48x - 36\)

1. Use the factor theorem to prove that \(x - 6\) is a factor of \(p(x)\). [2 marks]
2. 1. Prove that the graph of \(y = p(x)\) intersects the \(x\)-axis at exactly one point. [4 marks]
   2. State the coordinates of this point of intersection. [1 mark]

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

1. Use the factor theorem to show that \((x - 3)\) is a factor of \(f(x)\). [2]
2. Hence show that \(3\) is the only real root of the equation \(f(x) = 0\) [4]

Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right. [3 marks]