Graphical view of polynomial equations Problem set A

The graph of \(y = g(x)\), where \(g(x) = x^{4} - 3 x^{3} - 3 x^{2} + 16 x - 12\), is shown.

1. (a) Use the calculator to solve \(g(x) = 0\). Give the roots to 3 significant figures.
2. (b) One root is repeated. State it and say how the graph shows this.

The graphs of \(y = x^{3} + 2 x^{2} - 5 x\) and \(y = 2\) are shown. Solve \(x^{3} + 2 x^{2} - 5 x = 2\). Give the solutions to 3 significant figures.

(a) Write \(y = x^{2} + 12 x + 36\) in the form \(y = (x − p)^2 + q\). (b) State the coordinates of the vertex and explain why the equation \(x^{2} + 12 x + 36 = 0\) has one repeated root. Give the root.

The graph of \(y = x^3 − 3x^2 + c\) moves up or down as \(c\) changes. For each value of \(c\), state the number of different real roots of \(x^3 − 3x^2 + c = 0\).

1. (a) \(c = 6\)
2. (b) \(c = 4\)
3. (c) \(c = 2\)

The height of a roller-coaster track above the ground is \(h = t^3 − 9t^2 + 24t\) metres, where \(t\) is the time in seconds, \(0 \leq t \leq 6\). Find the times when the height is 18 metres.

A student draws the graphs of \(y = x^a − 4x + 1\) for \(a = 2, 3, 4\).

1. (a) Use the calculator to find how many real roots each graph has.
2. (b) Explain why a graph of \(y = x^3 + k\) always has at least one real root, but a graph of \(y = x^4 + k\) may have none.

The equation \(x^{3} - 12 x = k\) is solved by drawing \(y = x^{3} - 12 x\) and the line \(y = k\).

1. (a) Find the values of \(k\) for which the equation has three different solutions.
2. (b) Find the values of \(k\) for which the equation has exactly two different solutions.

The equation \(x^2 − 8x + k = 0\) has a repeated root.

1. (a) Find the value of \(k\) and the repeated root.
2. (b) Write down how many real roots \(x^2 − 8x + 13 = 0\) has. Explain using the graph.