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Polynomial Multiplication

Multiplying Polynomials Using the Distributive Property

Given two polynomials, their product can be calculated by using the Distributive Property. Consider, for example, the following pair of polynomials.
To multiply these two polynomials, the following four steps can be followed.
1
Distribute One Polynomial to All the Terms of the Other
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Start by writing the product
Next, distribute to each term of
2
Clear Parenthesis by Applying the Distributive Property
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Apply the Distributive Property one more time to clear all the parentheses.
3
Apply the Product of Powers Property
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Use the Product of Powers Property to rewrite some products as one single power.
4
Combine Like Terms and Simplify
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Finally, combine like terms and perform all the required operations to simplify the result.
Note that multiplying a polynomial with terms by a polynomial with terms produces products. Also, when two polynomials are multiplied, the product is a new polynomial whose degree equals the sum of the degrees of the multiplied polynomials.
Two polynomials can also be multiplied using the Box Method or the FOIL Method; however, the latter is useful only for multiplying binomials. These two methods are based on the Distributive Property.

Multiplying Polynomials Using the Box Method

Given two polynomials, their product can be calculated by using a box or table. Consider, for example, the following pair of polynomials.
To multiply these two polynomials, the following four steps can be followed.
1
Determine the Dimensions of the Table
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Start by drawing a table that has as many rows as there are terms in the first polynomial and that has as many columns as there are terms in the second polynomial.

Polynomial Number of Terms

For example, a table with rows and columns is needed to multiply by

2
Write the Row and Column Labels of the Table
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Now, write each term of the first polynomial at the left of each cell of the first column. Similarly, write each term of the second polynomial above each cell of the first row.

3
Multiply the Terms to Fill the Table
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Next, fill in the table's cells by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of and The remaining cells can be filled by following the same procedure.
4
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Finally, add all the expressions inside the table and combine like terms, if any.
The product of these polynomials has been found to be

The FOIL Method

The FOIL method is a mnemonic for remembering how to multiply two binomials. The word FOIL is an acronym for the words and Consider, for example, the following product.
These two binomials can be multiplied by following the next five steps.
1
Multiply the First Terms
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Start by multiplying the first terms of each binomial. In this case, multiply by
The empty box is there as a reminder that there are still missing terms.
2
Multiply the Outer Terms
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Next, multiply the outer terms, that is, multiply the first term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply by
3
Multiply the Inner Terms
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Now, multiply the inner terms — that is, multiply the second term of the left-hand side binomial by the first term of the right-hand side binomial. In this case, multiply by
4
Multiply the Last Terms
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Next, multiply the last terms of each binomial — that is, multiply the second term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply by
5
Simplify
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Finally, perform each product and combine like terms, if any, to simplify the resulting expression.
The following applet illustrates the FOIL method using two arbitrary binomials.
Like any other polynomial multiplication, the FOIL method is based on the Distributive Property.

Closure Property of Polynomial Multiplication

Given two polynomials and the product is always a polynomial.

Multiplying two polynomials produces a new polynomial.

In other words, the polynomials are closed under multiplication.

Proof

Consider two arbitrary polynomials and written in standard form.
These two polynomials can be multiplied by using the Distributive Property. The Product of Powers Property can also be applied to simplify the resulting expression.
Since and are polynomials, all the exponents are whole numbers. Furthermore, because the whole numbers are closed under addition, the exponents of the resulting expression are whole numbers. Then, the new expression can be rewritten as follows.
Consequently, the new expression is a polynomial. Therefore, the product of two polynomials produces a polynomial, which proves that the polynomials are closed under multiplication.