Find quotient and remainder by division - Factor & Remainder Theorem

10 marks Moderate -0.3

8. $\mathrm { p } ( x ) = x ^ { 4 } - ( x - 2 ) ^ { 4 }$.

Show that ( $x - 1$ ) is a factor of $\mathrm { p } ( x )$.
Show that $$\mathrm { p } ( x ) = 8 x ^ { 3 } - 24 x ^ { 2 } + 32 x - 16$$
Find the quotient and remainder when $\mathrm { p } ( x )$ is divided by ( $x + 1$ ).

12 marks Standard +0.3

7 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3$.

Find the quotient and remainder when $\mathrm { f } ( x )$ is divided by $( x + 1 )$.
Hence find the three roots of the equation $\mathrm { f } ( x ) = 0$. \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve $C$ with equation $y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9$.
Show that the $x$-coordinates of the stationary points on $C$ are given by $x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0$.
Use integration to find the exact area of the region enclosed by $C$ and the $x$-axis.

3 marks Easy -1.2

Let $f(x) = 8x^3 + 54x^2 - 17x - 21$.