Integration or area using factorised polynomial

10 marks Moderate -0.3

6 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30$.

Given that $x = 2$ is a root of the equation $\mathrm { f } ( x ) = 0$, express $\mathrm { f } ( x )$ as the product of 3 linear factors.
Use integration to find the exact value of $\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x$.
Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve $y = \mathrm { f } ( x )$ and the $x$-axis for $- 5 \leqslant x \leqslant 3$.

10 marks Standard +0.3

10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$

Use the factor theorem to show that $\mathrm { g } ( x )$ is divisible by $( x - 5 )$.
Hence, showing all your working, write $\mathrm { g } ( x )$ as a product of three linear factors. The finite region $R$ is bounded by the curve with equation $y = \mathrm { g } ( x )$ and the $x$-axis, and lies below the $x$-axis.
Find, using algebraic integration, the exact value of the area of $R$.

18 marks Moderate -0.8

6

The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6$.
1. Use the Factor Theorem to show that $x + 1$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6$ as the product of three linear factors.
The curve with equation $y = x ^ { 3 } - 7 x - 6$ is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the $x$-axis at the point $A$ and the points $B ( - 1,0 )$ and $C ( 3,0 )$.
1. State the coordinates of the point $A$.
2. Find $\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x$.
3. Hence find the area of the shaded region bounded by the curve $y = x ^ { 3 } - 7 x - 6$ and the $x$-axis between $B$ and $C$.
4. Find the gradient of the curve $y = x ^ { 3 } - 7 x - 6$ at the point $B$.
5. Hence find an equation of the normal to the curve at the point $B$.

17 marks Moderate -0.8

4

The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - x + 6$.
1. Find the remainder when $\mathrm { p } ( x )$ is divided by $x - 3$.
2. Use the Factor Theorem to show that $x + 2$ is a factor of $\mathrm { p } ( x )$.
3. Express $\mathrm { p } ( x ) = x ^ { 3 } - x + 6$ in the form $( x + 2 ) \left( x ^ { 2 } + b x + c \right)$, where $b$ and $c$ are integers.
4. The equation $\mathrm { p } ( x ) = 0$ has one root equal to - 2 . Show that the equation has no other real roots.
The curve with equation $y = x ^ { 3 } - x + 6$ is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the $x$-axis at the point $A ( - 2,0 )$ and the $y$-axis at the point $B$.
1. State the $y$-coordinate of the point $B$.
2. Find $\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x$.
3. Hence find the area of the shaded region bounded by the curve $y = x ^ { 3 } - x + 6$ and the line $A B$.

10 marks Moderate -0.3

$$g(x) = 2x^3 + x^2 - 41x - 70$$

Use the factor theorem to show that $g(x)$ is divisible by $(x - 5)$. [2]
Hence, showing all your working, write $g(x)$ as a product of three linear factors. [4]

The finite region $R$ is bounded by the curve with equation $y = g(x)$ and the $x$-axis, and lies below the $x$-axis.