Factorise polynomial completely

1. Factorise completely

$$x ^ { 3 } - 4 x ^ { 2 } + 3 x .$$

Factorise fully \(25 x - 9 x ^ { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-02_37_42_2700_1909}

5. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$

1. Using the factor theorem, explain why \(\mathrm { f } ( x )\) is divisible by \(( x + 3 )\).
2. Hence fully factorise \(\mathrm { f } ( x )\).
3. Show that \(\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 3 } + 5 x ^ { 2 } + 6 x }\) can be written in the form \(A + \frac { B } { x }\) where \(A\) and \(B\) are integers to be found.

4

1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
   1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
   2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
   3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
   1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
   2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
3. 1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
   2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.

5 Kaya is investigating the function $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 12 x + 45$$ Kaya makes two statements.
Statement 1: \(\mathrm { f } ( 3 ) = 0\) Statement 2: this shows that ( \(x + 3\) ) must be a factor of \(\mathrm { f } ( x )\).
5

1. State, with a reason, whether each of Kaya's statements is correct. Statement 1: \(\_\_\_\_\) Statement 2: \(\_\_\_\_\) 5
2. Fully factorise f (x).

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

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

The function f is a quartic function with real coefficients. The complex number \(5i\) is a root of the equation \(f(x) = 0\) Which one of the following must be a factor of \(f(x)\)? Circle your answer. [1 mark] \((x^2 - 25)\) \quad\quad \((x^2 - 5)\) \quad\quad \((x^2 + 5)\) \quad\quad \((x^2 + 25)\)

The cubic polynomial \(f(x)\) is given by \(f(x) = 2x^3 + ax^2 + bx + c\), where \(a\), \(b\), \(c\) are constants. The graph of \(f(x)\) intersects the \(x\)-axis at the points with coordinates \((-3, 0)\), \((2.5, 0)\) and \((4, 0)\). Find the coordinates of the point where the graph of \(f(x)\) intersects the \(y\)-axis. [5]