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

3. $$f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x - 6$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is - 5 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is no remainder.

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7 .

1. Show that \(a + b = 3\). When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is - 8 .
2. Find the value of \(a\) and the value of \(b\).

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants.

Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 7 ,

1. show that \(2 a - b = 6\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 ,
2. find the value of \(a\) and the value of \(b\).
   

2. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 4 x - 3\), where \(a\) and \(b\) are constants. Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a + b = 7$$ Given also that, when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is 9 ,
2. find the value of \(a\) and the value of \(b\), showing each step in your working.

2. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) the remainder is 25 ,

1. show that \(2 a + b = 5\) Given also that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_90_97_2613_1784} \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}

1. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 39 x + 20$$

1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise f ( \(x\) ) completely.

3. $$f ( x ) = ( 3 x - 2 ) ( x - k ) - 8$$ where \(k\) is a constant.

1. Write down the value of \(\mathrm { f } ( k )\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 4
2. Find the value of \(k\).
3. Factorise f(x) completely.

2. $$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 58 x + 40$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ). Given that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
2. find all the solutions of \(\mathrm { f } ( x ) = 0\).

1. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 5 x + 4$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\).
2. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Factorise f(x) completely.

4. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 10 x + 24$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise f(x) completely.

4. \(\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ) the remainder is 55
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 9

1. Find the value of \(a\) and the value of \(b\). Given that \(( 3 x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. factorise \(\mathrm { f } ( x )\) completely.

3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that \(a = - 9\)
2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.

4. \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + a x ^ { 2 } + 9 x - 18\), where \(a\) is a constant. Given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),

1. find the value of \(a\),
2. factorise \(\mathrm { f } ( x )\) completely,
3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).

2. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 4 x + 4$$

1. Use the factor theorem to show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\) completely.

3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,

1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(A\) and the value of \(B\).
3. Factorise f(x) fully.

4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
3. Factorise \(\mathrm { f } ( x )\) completely.

6. $$f ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 40 x + 21$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\)
2. Factorise f(x) completely.
3. Hence solve the equation $$6 \left( 2 ^ { 3 y } \right) + 7 \left( 2 ^ { 2 y } \right) = 40 \left( 2 ^ { y } \right) + 21$$ giving your answer to 2 decimal places.

3. $$f ( x ) = 24 x ^ { 3 } + A x ^ { 2 } - 3 x + B$$ where \(A\) and \(B\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 30

1. Show that \(A + 4 B = 114\) Given also that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find another equation in \(A\) and \(B\).
3. Find the value of \(A\) and the value of \(B\).
4. Hence find a quadratic factor of \(\mathrm { f } ( x )\).

7.

1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
2. Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
3. Prove that the value of \(x\) that satisfies $$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$ is a solution of the equation $$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$
4. State, with a reason, the value of \(x\) that satisfies equation (I).
   

3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$

1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ - 2 , - 1 ]\).
2. Find the coordinates of the turning point on the graph of \(y = \mathrm { f } ( x )\).
3. Given that \(\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)\), find the values of the constants, \(a , b\) and \(c\).
4. In the space provided on page 21, sketch the graph of \(y = \mathrm { f } ( x )\).
5. Hence sketch the graph of \(y = | \mathrm { f } ( x ) |\).

4. $$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$ where \(A\) and \(B\) are constants.
The complex number \(5 - \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(A\) and the value of \(B\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)

1. show that \(p = - 87\)
2. Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form. On an Argand diagram
   • the root \(z = 3\) is represented by the point \(P\)
   • the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
   • the number \(z = - 9\) is represented by the point \(S\)
   • Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
   • Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.

1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
(5)

1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.