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Fill in each blank so that the resulting statement is true.

“After performing polynomial long division, the answer may be checked by multiplying ____ by the ____ and then adding the ____. You should obtain the ____.”

This article aims to fill the blanks in the sentence. The article uses the concept of long division by the polynomial. A long division polynomial is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. Long division of polynomials also consists of a divisor, quotient, division, and remainder as in the long division method of numbers.

In algebra, the division of algebraic expressions can be done in these ways:

Dividing a monomial by another monomial.

Dividing a polynomial by a monomial.

Dividing a polynomial by a binomial.

Dividing a polynomial by another polynomial.

Steps for long division of polynomial

Here are the steps for the long division of polynomials:

Step 1. Arrange the members in descending order of their indices (if required). Write missing terms with zero as their coefficient.

Step 2. For the first term of the quotient, divide the dividend‘s first term by the divisor’s first term.

Step 3. Multiply this term of the quotient by the divisor to get the product.

Step 4. Subtract this product from the dividend and reduce the following term (if any). The difference and the reduced term will form a new dividend.

Step 5. Do this until you get a remainder, which can be zero or a lower index than the divisor.

Expert Answer

After a long division by a polynomial, the answer can be checked by multiplying the quotient by the divisor and then adding the remainder. You should get a dividend.

In algebra, long division by a polynomial is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of a well-known arithmetic technique called long division. This can easily be done manually as it breaks an otherwise complex division problem into smaller ones. Sometimes using a shortened version called synthetic division is faster, with less typing and less computation.

Numerical Result

After long division by a polynomial, the answer can be checked by multiplying the quotient by the divisor and then adding the remainder. You should get a dividend.

Example

Verify that the given statement is true or false:

“After a long division by a polynomial, the answer can be checked by dividing the quotient by divisor and then adding the remainder. You should get a dividend.”

Solution

The given statement is False. The correct statement is given as:

After a long division by a polynomial, the answer can be checked by multiplying the quotient by the divisor and then adding the remainder. It would help if you got a dividend.

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