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

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).

1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).

9 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. 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. Section B (36 marks)

8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).

9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h] \end{figure}

1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
   (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
   (B) Find the equation of the translated curve, simplifying your answer.

9

1. The polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$ Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\), and hence find the other two roots.
2. Hence solve the equation $$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$ for \(0 \leqslant x \leqslant 2 \pi\). Give each solution for \(x\) in an exact form.

6 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 3\) ) is a factor of \(\mathrm { f } ( x )\) and that, when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 35 .

1. Find the values of \(a\) and \(b\).
2. Using these values of \(a\) and \(b\), divide \(\mathrm { f } ( x )\) by ( \(x + 3\) ).

9 \includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-04_584_785_255_680} The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + 9 x ^ { 2 } + 10 x - 3\).

1. Verify that the curve crosses the \(x\)-axis at ( 3,0 ) and hence state a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Hence find the other two points of intersection of the curve with the \(x\)-axis.
4. The region enclosed by the curve and the \(x\)-axis is shaded in the diagram. Use integration to find the total area of this region.

5 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\).
2. Given that \(\mathrm { f } ( 2 ) = 0\), express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

9 The positive constant \(a\) is such that \(\int _ { a } ^ { 2 a } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 4 } { x ^ { 2 } } \mathrm {~d} x = 0\).

1. Show that \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\).
2. Show that \(a = 1\) is a root of \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\), and hence find the other possible value of \(a\), giving your answer in simplified surd form.

7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

1 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x - 14\), where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Using this value of \(a\), find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ).

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).

1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.

8 Two cubic polynomials are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + ( a - 3 ) x + 2 b , \quad \mathrm {~g} ( x ) = 3 x ^ { 3 } + x ^ { 2 } + 5 a x + 4 b$$ where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have a common factor of ( \(x - 2\) ), show that \(a = - 4\) and find the value of \(b\).
2. Using these values of \(a\) and \(b\), factorise \(\mathrm { f } ( x )\) fully. Hence show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have two common factors.

9 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ).
2. Show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.
3. Solve the equation $$4 \cos ^ { 3 } \theta - 7 \cos \theta - 3 = 0$$ for \(0 \leqslant \theta \leqslant 2 \pi\). Give each solution for \(\theta\) in an exact form.

7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30\).

1. Given that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = 0\), express \(\mathrm { f } ( x )\) as the product of 3 linear factors.
2. Use integration to find the exact value of \(\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\).
3. Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 5 \leqslant x \leqslant 3\).

3

1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.

7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).

1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
2. Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\). \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\).
3. Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
4. Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.

10

1. Use calculus to find, correct to 1 decimal place, the coordinates of the turning points of the curve \(y = x ^ { 3 } - 5 x\). [You need not determine the nature of the turning points.]
2. Find the coordinates of the points where the curve \(y = x ^ { 3 } - 5 x\) meets the axes and sketch the curve.
3. Find the equation of the tangent to the curve \(y = x ^ { 3 } - 5 x\) at the point \(( 1 , - 4 )\). Show that, where this tangent meets the curve again, the \(x\)-coordinate satisfies the equation $$x ^ { 3 } - 3 x + 2 = 0$$ Hence find the \(x\)-coordinate of the point where this tangent meets the curve again.

7

1. Show that
   (A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
   (B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\). Section B (36 marks)

2

1. Factorise fully \(n ^ { 3 } - n\).
2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .

1 Find the quotient and the remainder when \(x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1\) is divided by \(x ^ { 2 } + 5 x + 2\).

1 When the polynomial \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + 1\), the quotient is \(x ^ { 2 } + 4 x + 2\) and the remainder is \(x - 1\). Find \(\mathrm { f } ( x )\), simplifying your answer.