Verify factor then solve related equation - Factor & Remainder Theorem

6

Use the factor theorem to show that ( $x + 2$ ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$

4

The polynomial $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } - 4 x + 15$.
1. Use the Factor Theorem to show that $x + 3$ is a factor of $\mathrm { f } ( x )$.
2. Express $\mathrm { f } ( x )$ in the form $( x + 3 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
A curve has equation $y = x ^ { 4 } - 8 x ^ { 2 } + 60 x + 7$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Show that the $x$-coordinates of any stationary points of the curve satisfy the equation $$x ^ { 3 } - 4 x + 15 = 0$$
3. Use the results above to show that the only stationary point of the curve occurs when $x = - 3$.
4. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = - 3$.
5. Hence determine, with a reason, whether the curve has a maximum point or a minimum point when $x = - 3$.

2 marks

13

Given that $$P ( x ) = 125 x ^ { 3 } + 150 x ^ { 2 } + 55 x + 6$$ use the factor theorem to prove that ( $5 x + 1$ ) is a factor of $\mathrm { P } ( x )$.
[0pt] [2 marks]
13
Factorise $\mathrm { P } ( x )$ completely.
13
Hence, prove that $250 n ^ { 3 } + 300 n ^ { 2 } + 110 n + 12$ is a multiple of 12 when $n$ is a positive whole number.