3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
The degree of a polynomial is the highest exponent that appears among its terms. For example, $2 x ^ { 6 } - 3 x ^ { 2 } + 1$ is a polynomial of degree 6 .
(i) A polynomial $p ( x )$ has a turning point at ( 0,0 ). Explain why $p ( 0 ) = 0$ and why $p ^ { \prime } ( 0 ) = 0$, and explain why there is a polynomial $q ( x )$ such that
$$p ( x ) = x ^ { 2 } q ( x ) .$$
(ii) A polynomial $r ( x )$ has a turning point at ( $a , 0$ ) for some real number $a$. Write down an expression for $r ( x )$ that is of a similar form to the expression (*) above. Justify your answer in terms of a transformation of a graph.
(iii) You are now given that $f ( x )$ is a polynomial of degree 4 , and that it has two turning points at $( a , 0 )$ and at $( - a , 0 )$ for some positive number $a$.
(a) Write down the most general possible expression for $f ( x )$. Justify your answer.
(b) Describe a symmetry of the graph of $f ( x )$, and prove algebraically that $f ( x )$ does have this symmetry.
(c) Write down the $x$-coordinate of the third turning point of $f ( x )$.
(iv) Is there a polynomial of degree 4 which has turning points at $( 0,0 )$, at $( 1,3 )$, and at $( 2,0 )$ ? Justify your answer.
(v) Is there a polynomial of degree 4 which has turning points at ( 1,6 ), at ( 2,3 ), and at $( 4,6 )$ ? Justify your answer.
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A curve has equation
$$y = (2q - x^2)(2qx + 3)$$
The gradient of the curve at $x = -1$ is a function of $q$. Find the value of $q$ which minimises the gradient of the curve at $x = -1$.

The function f is defined for all real $x$ as
$$\mathrm{f}(x) = (p-x)(x+2)$$
Find the complete set of values of $p$ for which the maximum value of $\mathrm{f}(x)$ is less than 4.
A $-2 - 4\sqrt{2} < p < -2 + 4\sqrt{2}$
B $-2 - 2\sqrt{2} < p < -2 + 2\sqrt{2}$
C $-2\sqrt{5} < p < 2\sqrt{5}$
D $-6 < p < 2$
E $-4 < p < 0$
F $-2 < p < 2$

The minimum value of the function $x ^ { 4 } - p ^ { 2 } x ^ { 2 }$ is - 9 $p$ is a real number.
Find the minimum value of the function $x ^ { 2 } - p x + 6$
A - 3 B $6 - \frac { 3 \sqrt { 2 } } { 2 }$ C $\frac { 3 } { 2 }$ D 3 E $\frac { 9 } { 2 }$ F $6 + \frac { 3 \sqrt { 2 } } { 2 }$

A third-degree real-coefficient polynomial function $P(x)$ with leading coefficient 1 has two of its roots as $-5$ and $2$.
If $P(x)$ has a local extremum at the point $x = 0$, what is the third root?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 7 } { 3 }$
D) $\frac { -5 } { 2 }$
E) $\frac { -10 } { 3 }$

Let $a$ and $b$ be real numbers. It is known that the polynomial
$$f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 1$$
is

• increasing on the interval $( - \infty , 1 )$,
• decreasing on the interval $( 1,5 )$,
• increasing on the interval $( 5 , \infty )$.

Accordingly, what is $f ( 2 )$?
A) 0
B) 3
C) 6
D) 9
E) 12

Let $k$ and $m$ be real numbers. The functions $f$ and $g$ defined on the set of real numbers are
$$\begin{aligned} & f(x) = 2x^{3} - 9x^{2} - mx - k \\ & g(x) = x^{3} \cdot f(x) \end{aligned}$$
The functions $f$ and $g$ have local extrema at $x = -1$.
What is the sum $k + m$?
A) 31 B) 33 C) 35 D) 37 E) 39