Known polynomial, verify then factorise

6. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\).
3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)

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

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\) completely.
3. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$

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.

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.

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.

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.

7 In this question you must show detailed reasoning.

1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.

7

1. Use the factor theorem to show that \(( x - 2 )\) is a factor of \(x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
2. Factorise \(x ^ { 3 } + 6 x ^ { 2 } - x - 30\) completely.

1 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 13 x - 12\).

1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 3 x\).

1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
3. 1. Use the Remainder Theorem to find the remainder, \(r\), when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Using algebraic division, or otherwise, express \(\mathrm { p } ( x )\) in the form $$( x - 2 ) \left( x ^ { 2 } + a x + b \right) + r$$ where \(a , b\) and \(r\) are constants.

3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$

1. 1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Verify that \(\mathrm { p } ( 0 ) > \mathrm { p } ( 1 )\).
3. Sketch the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6\), indicating the values where the curve crosses the \(x\)-axis.

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]

\(f(x) = x^3 - 19x - 30\).

1. Show that \((x + 2)\) is a factor of \(f(x)\). [2]
2. Factorise \(f(x)\) completely. [4]

\(f(x) = x^3 + 4x^2 + x - 6\).

1. Use the factor theorem to show that \((x + 2)\) is a factor of \(f(x)\). [2]
2. Factorise \(f(x)\) completely. [4]
3. Write down all the solutions to the equation $$x^3 + 4x^2 + x - 6 = 0.$$ [1]