Algebra Answers

Collect like terms and simplify the expression:

12m2 – 9m + 5m – 4m2 – 7m + 10

sam began the week with a bag of buttons. at the end of the week he had used 2/3 of the buttons, and had 20 left how many did he start with at the beginning of the week

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ′(x) of the following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

Problem A.3

The formula for calculating the sum of all natural integers from 1 to n is well-known: Sn =1+2+3+...+n= n2 +n

 2

Similary, we know about the formula for calculating the sum of the first n squares:

n3 n2 n Qn =1·1+2·2+3·3+...+n·n= 3 + 2 + 6

Now, we reduce one of the two multipliers of each product by one to get the following sum: Mn =0·1+1·2+2·3+3·4+...+(n−1)·n

Find an explicit formula for calculating the sum Mn.

2. The functionsf and g are defined as below. f(x) = 3x+2: XER g(x)= 6 2x +3 Find the value of x for which f(g(x)) = 3 Sketch in a single diagram, the graphs of f(x) and f(x). Express each of f(x) and g(x), and solve the equation f¹(x) = g(x)

(a) What are quadratic residues and nonresidues?

(b) Write down the denition of elite primes in mathematical terms.

(c) Prove that

Q2t

i=0

Fi = 222t+1 􀀀 1.

(d) What does the  in the Brun-Titchmarsh inquality represent?

(e) Explicitly show that (x; 2t; 1)  x

2t .

(f) What can you say about the upper bound of E(x) for numbers of the form 32n + 1?

Find the smallest positive integer N that satisfies all of the following conditions:

 N is a square.

 N is a cube.

 N is an odd number.

 N is divisible by twelve different prime numbers.

How many digits does this number N have?

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + .... + n = (n2 + n)/2

Similary, we know about the formula for calculating the sum of the rst n squares:

Qn = 1 .1 + 2 . 2 + 3 .3 + ::: + n .n = n3/3 + n2/2 + n/6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 .1 + 1 . 2 + 2 . 3 + 3 . 4 + .....+ (n . 1) . n

Find an explicit formula for calculating the sum Mn.

The function is defined by f(x) = ax + b for x ER, where a and b are constants. It is given that f(2)= 1 and f(5) = 7. ii) Solve them and find the values of a and b. 1) Solve the equation f(f(x)) = 0. 1) Set up a pair of simultaneous equations using the information given.

The function is defined by ( ) for , where and are constants. It 

is given that ( ) and ( ) 

i) Set up a pair of simultaneous equations using the information given.

ii) Solve them and find the values of and .

iii) Solve the equation ( ( )) .