Question

In: Advanced Math

Using extended euclidean algorithm find f(x) and g(x) in: f(x)(x^5 + 4x^4 + 6x^3 + x^2...

Using extended euclidean algorithm find f(x) and g(x) in: f(x)(x^5 + 4x^4 + 6x^3 + x^2 + 4x + 6) + g(x)(x^5 + 5x^4 + 10x^3 + x^2 + 5x + 10) = x^3+1

Expert Solution

Lets denote

We have to find such that

Lets start with our division algorithm.

The last nonzero remainder is which can be written as linear combination of as follows:

(this is written based on the middle equation in the series of equations above)

(this is obtained by substituting the first equation in the series of equations above)

Simplifying, we get:

That is

So as to obtain the desired equation, dividing by 6:

Or

So we get and

Colby Messinger answered 2 years ago

a. Using the Euclidean Algorithm and Extended Euclidean Algorithm, show that gcd(99; 5) = 1 and...

a. Using the Euclidean Algorithm and Extended Euclidean Algorithm, show that gcd(99; 5) = 1 and find integers s1 and t1 such that 5s1 + 99t1 = 1. [Hint: You should find that 5(20) + 99(?1) = 1] b. Solve the congruence 5x 17 (mod 99) c. Using the Chinese Remainder Theorem, solve the congruence x 3 (mod 5) x 42 (mod 99) d. Using the Chinese Remainder Theorem, solve the congruence x 3 (mod 5) x 6 (mod 9)...

(a) For f(x) = 1 4 x 4 − 6x 2 find the intervals where f(x)...

(a) For f(x) = 1 4 x 4 − 6x 2 find the intervals where f(x) is concave up, and the intervals where f(x) is concave down, and the inflection points of f(x) by the following steps: i. Compute f 0 (x) and f 00(x). ii. Show that f 00(x) is equal to 0 only at x = −2 and x = 2. iii. Observe that f 00(x) is a continuous since it is a polynomial. Conclude that f 00(x)...

Given f''(x)= 4x-6 and f'(-2)=5 and f(-2)=1 FIND: Find f'(x)= and find f(2)=

Write a program( preferably in C++) using the extended Euclidean algorithm to find the multiplicative inverse of...

Write a program( preferably in C++) using the extended Euclidean algorithm to find the multiplicative inverse of a mod n. Your program should allow user to enter a and n. Note: For this question please make sure the code compiles and runs, it is not copied and pasted from elsewhere( I will be checking!). Thanks

Integral x^5-x^4+4x^3-4x^2+8x-4 / ( x^3+2)^3(x^2-1) dx

Given f(x) = x^4 - 4x^3 1) find the intervals on which f is increasing or...

Given f(x) = x^4 - 4x^3 1) find the intervals on which f is increasing or decreasing. 2) find the local maximum and minimum values of f. 3) find the intervals of concavity and the inflection points

a) Let f(x) = −x^4 − 4x^3 . (i) Find the intervals of increase/decrease of f....

a) Let f(x) = −x^4 − 4x^3 . (i) Find the intervals of increase/decrease of f. (ii) Find the local extrema of f (values and locations). (iii) Determine the intervals of concavity. (iv) Find the location of the inflection points of f. (v) Sketch the graph of f. (You can choose your own scale for the graph) b) A farmer wants to fence in an area of 6 km2 in a rectangular field and then divide it in half with...

For the function f(x) = x^2 +3x / 2x^2 + 6x +3 find the following, and...

For the function f(x) = x^2 +3x / 2x^2 + 6x +3 find the following, and use it to graph the function. Find: a)(2pts) Domain b)(2pts) Intercepts c)(2pts) Symmetry d) (2pts) Asymptotes e)(4pts) Intervals of Increase or decrease f) (2pts) Local maximum and local minimum values g)(4pts) Concavity and Points of inflection and h)(2pts) Sketch the curve

Let f(x) = −x^4 − 4x^3. (i) Find the intervals of increase/decrease of f. (ii) Find...

Let f(x) = −x^4 − 4x^3. (i) Find the intervals of increase/decrease of f. (ii) Find the local extrema of f (values and locations). (iii) Determine the intervals of concavity. (iv) Find the location of the inflection points of f. (v) Sketch the graph of f. (You can choose your own scale for the graph)

Let f(x) = x^4 - 4x^3 - 18 x^2 + 77 a) a) Find all critical values of the...

Let f(x) = x^4 - 4x^3 - 18 x^2 + 77 a) a) Find all critical values of the function. [10] b) b) Find all intervals of increase and decrease. [10] c) Find all relative extrema. Use the second derivative test. Label each as a relative max. or a relative min. [10] d) d) Find on what interval(s) the function is concave up and concave down. [10] e) e) Find all inflection point(s), if any, of the function. [10]