1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

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

1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
2. Sketch the graph of \(y = f(x)\). [3]
3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]

Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

The function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\). Find the values of \(b\) and \(c\). [5]

A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).

1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
2. Sketch the graph of \(y = f(x)\). [3]
3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]

You are given that \(\text{f}(x) = x^5 + kx - 20\). When \(\text{f}(x)\) is divided by \((x - 2)\), the remainder is 18. Find the value of \(k\). [3]

You are given that \(\text{f}(x) = (2x - 3)(x + 2)(x + 4)\).

1. Sketch the graph of \(y = \text{f}(x)\). [3]
2. State the roots of \(\text{f}(x - 2) = 0\). [2]
3. You are also given that \(\text{g}(x) = \text{f}(x) + 15\).
   1. Show that \(\text{g}(x) = 2x^3 + 9x^2 - 2x - 9\). [2]
   2. Show that \(\text{g}(1) = 0\) and hence factorise \(\text{g}(x)\) completely. [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$\text{f}'(x) = 6 - 4x - 3x^2,$$

1. find an expression for \(y\) in terms of \(x\), [5]
2. show that \(AB = k\sqrt{7}\), where \(k\) is an integer to be found. [6]

\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]

The line \(l\) is the tangent to \(C\) at \(O\).

1. Find an equation for \(l\). [4]
2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]

\(\text{f}(x) = x^3 - 6x^2 + 5x + 12\).

1. Show that $$(x + 1)(x - 3)(x - 4) \equiv x^3 - 6x^2 + 5x + 12.$$ [3]
2. Sketch the curve \(y = \text{f}(x)\), showing the coordinates of any points of intersection with the coordinate axes. [3]
3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
   1. \(y = \text{f}(x + 3)\),
   2. \(y = \text{f}(-x)\). [4]

\includegraphics{figure_8} The diagram shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]

The line \(l\) is the tangent to \(C\) at \(O\).

1. Find an equation for \(l\). [4]
2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]

$$\text{f}(x) = 4x - 3x^2 - x^3.$$

1. Fully factorise \(4x - 3x^2 - x^3\). [3]
2. Sketch the curve \(y = \text{f}(x)\), showing the coordinates of any points of intersection with the coordinate axes. [3]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]

A curve has the equation \(y = x^3 - 5x^2 + 7x\).

1. Show that the curve only crosses the \(x\)-axis at one point. [4]

The point \(P\) on the curve has coordinates \((3, 3)\).

1. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [6]

The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).

1. Show that triangle \(OQR\), where \(O\) is the origin, has area \(28\frac{1}{8}\). [3]

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]

\includegraphics{figure_1} 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]
2. Hence show that the equation of the curve may be written as \(y = 2x^3 + 11x^2 - x - 30\). [2]
3. Draw the line \(y = 5x + 10\) accurately on the graph. The curve and this line intersect at \((-2, 0)\); find graphically the \(x\)-coordinates of the other points of intersection. [3]
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2x^2 + 7x - 20 = 0.$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. [5]

\includegraphics{figure_3} Fig. 12 shows the graph of \(y = \frac{1}{x-3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{1}{x-3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]

You are given that \(n\), \(n + 1\) and \(n + 2\) are three consecutive integers.

1. Expand and simplify \(n^2 + (n + 1)^2 + (n + 2)^2\). [2]
2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. [2]