Question

Whole numbers are closed under addition because the sum of two whole numbers is always a whole number. Explain how the process of checking polynomial division supports the fact that polynomials are closed under multiplication and addition.

Answer 1

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When checking polynomial division you multiply the quotient by the divisor..

The quotient will be a polynomial (with or without a remainder). multiplying this polynomial by the polynomial divisor. We get a polynomial in which the exponennts and coefficients have changed thus polynomial are closed under multiplication

There will usually be at least two terms in the divisor..when we multiply the quotient by this, we will use the distributive property, multiplying the entire quotient by each term of the divisor. When this process is finished. We will need to add the polynomials we had from multiplying together..doing this we get a polynomial answer in which the coefficients have changed..this polynomial are closed under addition..

Answer 2

Answer:

The quotient is a polynomial (with or without a remainder). multiplying this polynomial by the divisor of the polynomial We receive a polynomial with modified exponents and coefficients, therefore the polynomial is closed under multiplication.

• If there is no remainder, the dividend equals the quotient multiplied by the divisor.

• The quotient and divisor are both polynomials, as is their product, the dividend.

• If a residual exists, it is added to the product of the quotient and the divisor.

• The dividend, which is a polynomial, is the sum of the remainder and the product of the quotient and divisor.

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