Question

What is the remainder obtained by dividing x7 x5 1 by the generator polynomial x3 1?

Answer 1

The remainder obtained by dividing on ${x^7} + {x^5} + 1$ the generator polynomial ${x^3} + 1$ is $\boxed{{-x^2} + {x} +1}$

Given:

The generator polynomial is ${x^3} + 1.$

The polynomial is ${x^7} + {x^5} + 1$

Explanation:

There are two methods of dividing the two polynomials.

(1). Synthetic division

(2). Long division method

The fraction can be expressed as,

${\text{Fraction}}= \dfrac{{{x^7} + {x^5} + 1}}{{{x^3} + 1}}$

The denominator of the fraction is ${x^3} + 1$ and the numerator of the fraction is ${x^7} + {x^5} + 1.$

Divide ${x^7} + {x^5} + 1$ by ${x^3} + 1$ to obtain the remainder and the quotient of the polynomial.

The remainder obtained by dividing on ${x^7} + {x^5} + 1$ the generator polynomial ${x^3} + 1$ is $\boxed{{-x^2} + {x} +1}$

Kindly refer the image attached below.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Long division method

Keywords: division, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree, cubic polynomial, quadratic equation.

Answer 2

Please, use " ^ " to indicate exponention.  x7 and  x3 are meaningless.

I must assume you mean x^7 + 0 x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 1
and that your "generator polynomial" is   x^3 + 0x^2 + 0x + 1.  If this is not the case, then you really need to work on your presentation of polynomials.


Divide (x^3 + 0x^2 + 0x + 1 into 
     x^7 + 0 x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 1

Dividing x^3 into x^7 leaves a partial quotient of x^4.  Multiply (x^3 + 1) by this x^4 and write your result under x^7 + 0x^6 + 0x^5  ...  + 1
Subtract.

Can you now finish this problem?  If not, what do you need to know so that you can finish it?