Graphical view of polynomial equations Problem set B Worked solutions

The graph of \(y = g(x)\), where \(g(x) = x^{4} - 7 x^{2} + 10 x - 4\), is shown.

1. (a) Use the calculator to solve \(g(x) = 0\). Give the roots to 3 significant figures.

   • Graph \(y = x^{4} - 7 x^{2} + 10 x - 4\) and find the \(x\)-intercepts.
   • From the calculator: \(x = −3.24, 1.00, 1.24\)

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

   • At \(x = 1\) the curve touches the axis and does not cross it.
   • So \((x − 1)\) is a factor twice: \(x = 1\) is a repeated root.

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

• Solutions of \(f(x) = −3\) are the \(x\)-values where the curve meets the line \(y = −3\).
• From the calculator: intersections at \(x = −1.13, 0.798, 3.33\)
• Check: moving −3 across gives \(x^{3} - 3 x^{2} - 2 x + 3 = 0\), with the same three roots.

(a) Write \(y = x^{2} - 10 x + 25\) in the form \(y = (x − p)^2 + q\). (b) State the coordinates of the vertex and explain why the equation \(x^{2} - 10 x + 25 = 0\) has one repeated root. Give the root.

• Half of −10 is −5, and \((−5)^2 = 25\), so \(x^{2} - 10 x + 25 = (x − 5)^2 + 0\).
• The vertex is \((5, 0)\), on the \(x\)-axis.
• The parabola touches the axis at one point: \(x = 5\) (repeated).

The graph of \(y = x^3 − 12x + c\) moves up or down as \(c\) changes. For each value of \(c\), state the number of different real roots of \(x^3 − 12x + c = 0\).

1. (a) \(c = 20\)

   • Graph \(y = x^{3} - 12 x + 20\).
   • From the graph it meets the \(x\)-axis at 1 point.

2. (b) \(c = 16\)

   • Graph \(y = x^{3} - 12 x + 16\).
   • From the graph it meets the \(x\)-axis at 2 points; one of them is a touch (repeated root).

3. (c) \(c = 5\)

   • Graph \(y = x^{3} - 12 x + 5\).
   • From the graph it meets the \(x\)-axis at 3 points.

The speed of a cyclist is \(v = t^3 − 12t^2 + 36t\) km/h, where \(t\) is the time in minutes, \(0 \leq t \leq 8\). Find the times when the speed is 10 km/h.

• Graph \(y = x^{3} - 12 x^{2} + 36 x\) and the line \(y = 10\).
• From the calculator: intersections at \(t = 0.309, 4.51, 7.18\)
• All three values lie in the time allowed, so all three are answers.

Three graphs are drawn: \(y = x^a − 2x + 2\) for \(a = 2, 3, 4\).

1. (a) Use the calculator to find how many real roots each graph has.

   • Graph each of \(y = x^2 − 2x + 2\), \(y = x^3 − 2x + 2\), \(y = x^4 − 2x + 2\).
   • Count the \(x\)-intercepts: 0, 1, 0.

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 \(x^{3} - 3 x^{2} - 9 x = k\) is solved by drawing \(y = x^{3} - 3 x^{2} - 9 x\) and the line \(y = k\).

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

   • Turning points of \(y = x^{3} − 3 x^{2} − 9 x\): from the calculator, a minimum at height \(−27\) and a maximum at height \(5\).
   • The line meets the curve three times when it is between these heights: \(−27 < k < 5\).

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 \(k = −27\) or \(k = 5\).
   • Those are the two heights that give a repeated root.

The equation \(x^2 + 6x + k = 0\) has a repeated root.

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

   • Complete the square: \(x^2 + 6x + k = (x + 3)^2 + k − 9\).
   • A repeated root needs the vertex on the axis: \(k − 9 = 0\), so \(k = 9\).
   • The root is \(x = −3\).

2. (b) Write down how many real roots \(x^2 + 6x + 11 = 0\) has. Explain using the graph.

   • Changing \(k\) to \(11\) moves the graph up by 2.
   • The vertex is now at height 2, above the axis.
   • So the graph does not meet the axis: 0 real roots.