Factorise polynomial completely

1. Factorise completely

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

Factorise fully \(25 x - 9 x ^ { 3 }\)
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1 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
\(2 \quad\) (i) Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
(ii) Factorise \(x ^ { 2 } - 4 \quad x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.

12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$

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.
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   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.

4. The polynomial \(P ( x )\) is defined as follows:
\(P ( x ) \equiv x ^ { 8 } + 8 x ^ { 7 } + 28 x ^ { 6 } + 56 x ^ { 5 } + 70 x ^ { 4 } + 56 x ^ { 3 } + 28 x ^ { 2 } + 8 x , x \in \mathbb { R }\)
By first factorising \(P ( x )\) find all of its real roots.
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12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).

1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
3. Express \(\mathrm { f } ( x )\) in fully factorised form.
4. Sketch the graph of \(y = \mathrm { f } ( x )\).
5. Solve the equation \(\mathrm { f } ( x ) = 12\).

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