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

\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). The values of \(x\) where it crosses the \(x\)-axis are \(-5\), \(-2\) and \(2\), and it crosses the \(y\)-axis at \((0, -20)\).

1. Express f(\(x\)) in factorised form. [2]
2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4. Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place. [6]
4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]

Fig. 9 shows a sketch of the curve \(y = x^3 - 3x^2 - 22x + 24\) and the line \(y = 6x + 24\). \includegraphics{figure_9}

1. Differentiate \(y = x^3 - 3x^2 - 22x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places. [4]
2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = -4\). Find algebraically the \(x\)-coordinate of the other point of intersection. [3]
3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6x + 24\) for \(-4 \leq x \leq 0\), shown shaded on Fig. 9. [4]

\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A\), \(B\) and \(C\). Given that \(A\) has coordinates \((-4, 0)\),

1. show that \(k = -7\), [2]
2. find the coordinates of \(B\) and \(C\). [5]

\(f(x) = 2x^3 + 3x^2 - 6x + 1\).

1. Find the remainder when \(f(x)\) is divided by \((2x - 1)\). [2]
2. 1. Find the remainder when \(f(x)\) is divided by \((x + 2)\).
   2. Hence, or otherwise, solve the equation $$2x^3 + 3x^2 - 6x - 8 = 0,$$ giving your answers to 2 decimal places where appropriate. [7]

\includegraphics{figure_2} Figure 2 shows the curves with equations \(y = 7 - 2x - 3x^2\) and \(y = \frac{2}{x}\). The two curves intersect at the points \(P\), \(Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P\), \(Q\) and \(R\) satisfy the equation $$3x^3 + 2x^2 - 7x + 2 = 0.$$ [2] Given that \(P\) has coordinates \((-2, -1)\),
2. find the coordinates of \(Q\) and \(R\). [6]

$$f(x) = 2x^3 - 5x^2 + x + 2.$$

1. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
2. Fully factorise \(f(x)\). [4]
3. Solve the equation \(f(x) = 0\). [1]
4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2\pi\) for which $$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$ giving your answers in terms of \(\pi\). [4]

$$f(x) = x^3 + kx - 20.$$ Given that f(x) is exactly divisible by \((x + 1)\),

1. find the value of the constant \(k\), [2]
2. solve the equation \(f(x) = 0\). [4]

The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.

1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]

Using \(x = 6\),

1. find the common ratio of the series, [1]
2. find the sum of the first eight terms of the series. [2]

The polynomial f(x) is given by $$\text{f}(x) = x^3 + kx^2 - 7x - 15,$$ where \(k\) is a constant. When f(x) is divided by \((x + 1)\) the remainder is \(r\). When f(x) is divided by \((x - 3)\) the remainder is \(3r\).

1. Find the value of \(k\). [5]
2. Find the value of \(r\). [1]
3. Show that \((x - 5)\) is a factor of f(x). [2]
4. Show that there is only one real solution to the equation f(x) = 0. [4]

\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).

1. Write down \(\frac{dy}{dx}\). [2]
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]

The functions f and g are defined with their respective domains by $$f(x) = x^2 \quad \text{for all real values of } x$$ $$g(x) = \frac{1}{2x + 1} \quad \text{for real values of } x, \quad x \neq -0.5$$

1. Explain why f does not have an inverse. [1]
2. The inverse of g is \(g^{-1}\). Find \(g^{-1}(x)\). [3]
3. State the range of \(g^{-1}\). [1]
4. Solve the equation \(fg(x) = g(x)\). [3]

The sequence defined by $$x_1 = 3, \quad x_{n+1} = \sqrt{31 - \frac{5}{2}x_n}$$ converges to the number \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration. [3]
2. Find an equation of the form \(ax^3 + bx + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root. [3]

You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]

Find the quotient and the remainder when \(x^4 + 3x^3 + 5x^2 + 4x - 1\) is divided by \(x^2 + x + 1\). [4]

Find the quotient and remainder when \((x^4 + x^3 - 5x^2 - 9)\) is divided by \((x^2 + x - 6)\). [4]

Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]

Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]

A student was attempting to prove that \(x = \frac{1}{2}\) is the only real root of $$x^3 + \frac{1}{4}x - \frac{1}{2} = 0.$$ The attempted solution was as follows. $$x^3 + \frac{1}{4}x = \frac{1}{2}$$ $$\therefore \quad x(x^2 + \frac{1}{4}) = \frac{1}{2}$$ $$\therefore \quad x = \frac{1}{2}$$ or $$x^2 + \frac{1}{4} = \frac{1}{2}$$ i.e. $$x^2 = -\frac{1}{4} \quad \text{no solution}$$ $$\therefore \quad \text{only real root is } x = \frac{1}{2}$$

1. Explain clearly the error in the above attempt. [2]
2. Give a correct proof that \(x = \frac{1}{2}\) is the only real root of \(x^3 + \frac{1}{4}x - \frac{1}{2} = 0\). [3]

The equation $$x^3 + \beta x - \alpha = 0 \quad \text{(I)}$$ where \(\alpha\), \(\beta\) are real, \(\alpha \neq 0\), has a real root at \(x = \alpha\).

1. Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(|\alpha| < 2\). [6]

An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation (I) but the incorrect method used by the student produces 3 distinct real "roots".

1. Find the range of possible values for \(\alpha\). [7]

$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]

Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,

1. find the set of values that \(p\) can take. [5]

The cubic polynomial \(\text{f}(x)\) is defined by \(\text{f}(x) = x^3 + px + q\), where \(p\) and \(q\) are constants.

1. 1. Given that \(\text{f}'(2) = 13\), find the value of \(p\). [2]
   2. Given also that \((x - 2)\) is a factor of \(\text{f}(x)\), find the value of \(q\). [2]
   The curve \(y = \text{f}(x)\) is translated by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\).
2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]