Sketching Polynomial Curves

13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Find the other factors of \(\mathrm { f } ( x )\).
3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
   Solve this equation to find the \(x\)-coordinates of A and B .

2 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).

1. Use the intersections with both axes to express the equation of the curve in a factorised form.
2. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
3. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection.

1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$

4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).

3 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).

1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
3. Express \(\mathrm { f } ( x )\) in fully factorised form.
4. Sketch the graph of \(y = \mathrm { f } ( x )\).
5. Solve the equation \(\mathrm { f } ( x ) = 12\).

9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$

1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.

12 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).

1. (a) Factorise completely \(x ^ { 3 } + 10 x ^ { 2 } + 25 x\) (b) Sketch the curve with equation

$$y = x ^ { 3 } + 10 x ^ { 2 } + 25 x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. The point with coordinates \(( - 3,0 )\) lies on the curve with equation $$y = ( x + a ) ^ { 3 } + 10 ( x + a ) ^ { 2 } + 25 ( x + a )$$ where \(a\) is a constant.
(c) Find the two possible values of \(a\).

2 Two curves have equations \(y = \ln x\) and \(y = \frac { k } { x }\), where \(k\) is a positive constant.

1. Sketch the curves on a single diagram.
2. Explain how your diagram shows that the equation \(x \ln x - k = 0\) has exactly one real root.