please do all $4$ of the problems. these are computer science related problems.

What is the minimum number of multiplications that you need to calculate the

monomial x$^(39)$ for the given value x $?$

Ans:

We are given a polynomial with the following structure

P$($x$)=($a$\_(0)+$a$\_(1)$x$+$a$\_(2)$x$^(2)+$a$\_(3)$x$^(3))($b$\_(0)+$b$\_(1)$x$+$b$\_(2)$x$^(2))($c$\_(0)+$c$\_(1)$x$+$c$\_(2)$x$^(2)+$c$\_(3)$x$^(3)+$c$\_(4)$x$^(4)),$

where a$\_(0),$a$\_(1),$a$\_(2),$a$\_(3),$b$\_(0),$b$\_(1),$b$\_(2),$c$\_(0),$c$\_(1),$c$\_(2),$c$\_(3),$ and c$\_(4)$ are arbitrarily given coefficients.

What is the minimum number of multiplications do you need to find the value

of P$($x$)$ for any value of x $?$

Ans:

We are given a polynomial with the following structure

P$($x$)=($a$\_(0)+$a$\_(1)$x$+$a$\_(2)$x$^(2)+$a$\_(3)$x$^(3))^(4)($b$\_(0)+$b$\_(1)$x$+$b$\_(2)$x$^(2))^(2)($c$\_(0)+$c$\_(1)$x$+$c$\_(2)$x$^(2)+$c$\_(3)$x$^(3)+$c$\_(4)$x$^(4))^(8),$

where a$\_(0),$a$\_(1),$a$\_(2),$a$\_(3),$b$\_(0),$b$\_(1),$b$\_(2),$c$\_(0),$c$\_(1),$c$\_(2),$c$\_(3),$ and c$\_(4)$ are arbitrarily given coefficients.

What is the minimum number of multiplications do you need to find the value

of P$($x$)$ for any value of x $?$

Ans:

Solve the first order linear recurrence relation: E$\_($n$)=3$E$\_($n$-1)+2$ with the initial value

E$\_(0)=0.$

Solution: