Graphical view of polynomial equations Problem set A Worked solutions

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.

   • Graph \(y = x^{4} - 3 x^{3} - 3 x^{2} + 16 x - 12\) and find the \(x\)-intercepts.
   • From the calculator: \(x = −2.30, 1.30, 2.00\)

2. (b) One root is repeated. State it and say how the graph shows this.

   • At \(x = 2\) the curve touches the axis and does not cross it.
   • So \((x − 2)\) is a factor twice: \(x = 2\) is a repeated root.

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.

• Solutions of \(f(x) = 2\) are the \(x\)-values where the curve meets the line \(y = 2\).
• From the calculator: intersections at \(x = −3.32, −0.358, 1.68\)
• Check: moving 2 across gives \(x^{3} + 2 x^{2} - 5 x - 2 = 0\), with the same three roots.

(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.

• Half of 12 is 6, and \(6^2 = 36\), so \(x^{2} + 12 x + 36 = (x + 6)^2 + 0\).
• The vertex is \((−6, 0)\), on the \(x\)-axis.
• The parabola touches the axis at one point: \(x = −6\) (repeated).

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) \(c = 6\)

   • Graph \(y = x^{3} - 3 x^{2} + 6\).
   • From the graph it meets the \(x\)-axis at 1 point.

2. (b) \(c = 4\)

   • Graph \(y = x^{3} - 3 x^{2} + 4\).
   • From the graph it meets the \(x\)-axis at 2 points; one of them is a touch (repeated root).

3. (c) \(c = 2\)

   • Graph \(y = x^{3} - 3 x^{2} + 2\).
   • From the graph it meets the \(x\)-axis at 3 points.

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.

• Graph \(y = x^{3} - 9 x^{2} + 24 x\) and the line \(y = 18\).
• From the calculator: intersections at \(t = 1.27, 3.00, 4.73\)
• All three values lie in the time allowed, so all three are answers.

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.

   • Graph each of \(y = x^2 − 4x + 1\), \(y = x^3 − 4x + 1\), \(y = x^4 − 4x + 1\).
   • Count the \(x\)-intercepts: 2, 3, 2.

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.

   • For \(x^3 + k\) one end goes to \(−\infty\) and the other to \(+\infty\), so the graph crosses the axis at least once.
   • For \(x^4 + k\) both ends go to \(+\infty\). If \(k > 0\) the whole graph is above the axis: no real roots.

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.

   • Turning points of \(y = x^{3} − 12 x\): from the calculator, a minimum at height \(−16\) and a maximum at height \(16\).
   • The line meets the curve three times when it is between these heights: \(−16 < k < 16\).

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

   • The line touches the curve at a turning point and cuts it once more when \(k = −16\) or \(k = 16\).
   • Those are the two heights that give a repeated root.

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

1. (a) Find the value of \(k\) and the repeated root.

   • Complete the square: \(x^2 − 8x + k = (x − 4)^2 + k − 16\).
   • A repeated root needs the vertex on the axis: \(k − 16 = 0\), so \(k = 16\).
   • The root is \(x = 4\).

2. (b) Write down how many real roots \(x^2 − 8x + 13 = 0\) has. Explain using the graph.

   • Changing \(k\) to \(13\) moves the graph down by 3.
   • The vertex is now at height −3, below the axis.
   • So the graph cuts the axis: 2 real roots.