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

7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.

1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.

3

1. Find the value of \(a\) for which ( \(x - 2\) ) is a factor of \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8\).
2. Show that, for this value of \(a\), the cubic equation \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8 = 0\) has only one real root.

9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are real, is denoted by \(\mathrm { p } ( x )\).

1. Give a reason why the equation \(\mathrm { p } ( x ) = 0\) has at least one real root.
2. Suppose that the curve with equation \(y = \mathrm { p } ( x )\) has a local minimum point and a local maximum point with \(y\)-coordinates \(y _ { \text {min } }\) and \(y _ { \text {max } }\) respectively.
   1. Prove that if \(y _ { \text {min } } y _ { \text {max } } < 0\), then the equation \(\mathrm { p } ( x ) = 0\) has three real roots.
   2. Comment on the number of distinct real roots of the equation \(\mathrm { p } ( x ) = 0\) in the case \(y _ { \text {min } } y _ { \text {max } } = 0\).
   3. Suppose instead that the equation \(\mathrm { p } ( x ) = 0\) has only one real root for all values of \(c\). Prove that \(a ^ { 2 } \leqslant 3 b\).
   4. The iterative scheme $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 1 } { 3 x _ { n } ^ { 2 } + 1 } , \quad x _ { 0 } = 0$$ converges to a root of the cubic equation \(\mathrm { p } ( x ) = 0\).
      (a) Find \(\mathrm { p } ( x )\).
      (b) Find the limit of the iteration, correct to 4 decimal places.
   5. Determine the rate of convergence of the iterative scheme.

Given that \(( x - 2 )\) and \(( x + 2 )\) are factors of the polynomial \(2 x ^ { 3 } + p x ^ { 2 } + q x - 12\), a) find the values of \(p\) and \(q\),
b) determine the other factor of the polynomial.

Solve the equation $$\frac { 6 x ^ { 5 } - 17 x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } } { ( 3 x + 2 ) } = 0$$

\includegraphics{figure_3} The diagram shows the curve \(y = 2x^5 + 3x^3\) and the line \(y = 2x\) intersecting at points \(A\), \(O\) and \(B\).

1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2x^4 + 3x^2 - 2 = 0\). [2]
2. Solve the equation \(2x^4 + 3x^2 - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form. [3]

The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).

1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 - ax^2 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 2)\) is a factor of \(p(x)\), and that the remainder is 35 when \(p(x)\) is divided by \((x - 3)\).

1. Find the values of \(a\) and \(b\). [5]
2. Hence factorise \(p(x)\) and show that the equation \(p(x) = 0\) has exactly one real root. [3]

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = 6x^3 + ax^2 + 3x - 10,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

1. Find the value of \(a\) and hence factorise \(\mathrm{p}(x)\) completely. [5]
2. Solve the equation \(\mathrm{p}(\cos\theta) = 0\) for \(-90° < \theta < 90°\). [2]

The polynomial \(\text{p}(x)\) is defined by $$\text{p}(x) = ax^3 - ax^2 - 15x + 18,$$ where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(\text{p}(x)\).

1. Find the value of \(a\). [2]
2. Hence factorise \(\text{p}(x)\) completely. [3]

The polynomial \(x^4 - 6x^2 + x + a\) is denoted by \(f(x)\).

1. It is given that \((x + 1)\) is a factor of \(f(x)\). Find the value of \(a\). [2]
2. When \(a\) has this value, verify that \((x - 2)\) is also a factor of \(f(x)\) and hence factorise \(f(x)\) completely. [4]

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 + 3x^2 + bx + 12,$$ where \(a\) and \(b\) are constants. It is given that \((x + 3)\) is a factor of \(p(x)\). It is also given that the remainder is 18 when \(p(x)\) is divided by \((x + 2)\).

1. Find the values of \(a\) and \(b\). [5]
2. When \(a\) and \(b\) have these values,
   1. show that the equation \(p(x) = 0\) has exactly one real root, [4]
   2. solve the equation \(p(\sec y) = 0\) for \(-180° < y < 180°\). [3]

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

1. Use the factor theorem to find the value of \(a\). [2]
2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]

\includegraphics{figure_4} The diagram shows the curve with equation $$y = x^4 + 2x^3 + 2x^2 - 12x - 32.$$ The curve crosses the \(x\)-axis at points with coordinates \((\alpha, 0)\) and \((\beta, 0)\).

1. Use the factor theorem to show that \((x + 2)\) is a factor of $$x^4 + 2x^3 + 2x^2 - 12x - 32.$$ [2]
2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt[3]{p + qx}\), and state the values of \(p\) and \(q\). [3]
3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]

Let \(f(x) = 8x^3 + 54x^2 - 17x - 21\).

1. Show that \(x + 7\) is a factor of \(f(x)\). [1]
2. Find the quotient when \(f(x)\) is divided by \(x + 7\). [2]

The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by p\((x)\). It is given that \((x + 2)\) is a factor of p\((x)\) and that when p\((x)\) is divided by \((x + 1)\) the remainder is 2. Find the values of \(a\) and \(b\). [5]

The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(\text{p}(x)\). It is given that \((3x + 1)\) is a factor of \(\text{p}(x)\).

1. Find the value of \(a\). [3]
2. When \(a\) has this value, factorise \(\text{p}(x)\) completely. [3]

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

Factorise completely $$x^3 - 4x^2 + 3x.$$ [3]

Given that \(f(x) = (x^2 - 6x)(x - 2) + 3x\),

1. express \(f(x)\) in the form \(a(x^2 + bx + c)\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Hence factorise \(f(x)\) completely. [2]
3. Sketch the graph of \(y = f(x)\), showing the coordinates of each point at which the graph meets the axes. [3]

$$f(x) = Ax^3 + 6x^2 - 4x + B$$ where \(A\) and \(B\) are constants. Given that

• \((x + 2)\) is a factor of \(f(x)\)
• \(\int_{-3}^{5} f(x)dx = 176\)

Find the value of \(A\) and the value of \(B\). [7]

\(f(x) = x^3 - 2x^2 + ax + b\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \((x - 2)\), the remainder is 1. When \(f(x)\) is divided by \((x + 1)\), the remainder is 28.

1. Find the value of \(a\) and the value of \(b\). [6]
2. Show that \((x - 3)\) is a factor of \(f(x)\). [2]