# Find Factors of Polynomial Division: x-2 in Q[x] & x+1 in Z5[x]

• dash
In summary, polynomial division is the process of dividing one polynomial by another, similar to long division in arithmetic. To find the values of k for which x-2 and x+1 are factors, one must perform the division and solve for k. It is important to show effort and attempt the problem before seeking help.
dash
1. Polynomial division

a) For what values of k is x-2 a factor x^4 – 5x^3 + 3x + k in Q[x]?

b) For what values of k is x+1 a factor of x^4 + 2x^3 – 3x^2 + kx + 1 in Z5[x]

Dash, you need to show what effort you have put into the problem before we can help you. This is both common sense and site policy. Until you are willing to do so, please stop posting your homework questions.

my problem is that I don't know how to start the problem?

Are you saying you do not know how to divide? "Polynomial division" is basically the same as "long division" in arithmetic: choose a "trial quotient" based on the term with the highest power, multiply by the divisor and subtract from the dividend. Repeat.

## 1. What is polynomial division?

Polynomial division is a mathematical process used to divide one polynomial by another polynomial. It is similar to long division, but instead of dividing numbers, we are dividing terms in a polynomial expression.

## 2. What are factors in polynomial division?

In polynomial division, factors refer to the terms that can be multiplied together to obtain the original polynomial. In other words, they are the numbers or expressions that divide evenly into the polynomial without any remainder.

## 3. How do I find the factors of a polynomial in Q[x]?

To find the factors of a polynomial in Q[x] (rational coefficients), you can use the rational root theorem. This theorem states that if a polynomial has rational roots, they will be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By testing these possible roots, we can find the factors of the polynomial.

## 4. How do I find the factors of a polynomial in Z5[x]?

In Z5[x] (polynomials with coefficients in the ring of integers mod 5), we can use the same method as in Q[x], but we only need to test the possible roots 0, 1, 2, 3, and 4 since these are the only integers in Z5. Additionally, we can also use the factor theorem, which states that if a polynomial has a root r, then (x-r) is a factor of the polynomial.

## 5. How do I use the factors to divide a polynomial in polynomial division?

To divide a polynomial, we use the factors we have found to perform long division. We start by dividing the highest degree term of the polynomial by the highest degree factor, and then continue with the remainder of the polynomial. This process is repeated until there is no remainder, and the quotient is the result of the polynomial division.

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