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Reference

Polynomial Multiplication

Method

Multiplying Polynomials Using the Distributive Property

The product of two polynomials can be calculated by using the Distributive Property. Consider, for example, the following pair of polynomials.

P (x) Q (x) = x^{3} + 2 x^{2} - 3 = x^{2} + 4

To multiply these two polynomials, the following four steps can be followed.

Distribute One Polynomial to All the Terms of the Other

Start by writing the product

P (x) \cdot Q (x) .

(x^{3} + 2 x^{2} - 3) (x^{2} + 4)

Next, distribute

P (x)

to each term of

Q (x) .

(x^{3} + 2 x^{2} - 3) (x^{2} + 4)

Distr

Distribute $x^{3} + 2 x^{2} - 3$

(x^{3} + 2 x^{2} - 3) x^{2} + (x^{3} + 2 x^{2} - 3) 4

Clear Parentheses by Applying the Distributive Property

Apply the Distributive Property again to clear all the parentheses.

(x^{3} + 2 x^{2} - 3) x^{2} + (x^{3} + 2 x^{2} - 3) 4

Distr

Distribute $x^{2}$

(x^{3} \cdot x^{2} + 2 x^{2} \cdot x^{2} - 3 \cdot x^{2}) + (x^{3} + 2 x^{2} - 3) 4

Distr

Distribute $4$

(x^{3} \cdot x^{2} + 2 x^{2} \cdot x^{2} - 3 \cdot x^{2}) + (x^{3} \cdot 4 + 2 x^{2} \cdot 4 - 3 \cdot 4)

Apply the Product of Powers Property

Use the Product of Powers Property to rewrite the product of two or more exponents as a single power.

(x^{3} \cdot x^{2} + 2 x^{2} \cdot x^{2} - 3 \cdot x^{2}) + (x^{3} \cdot 4 + 2 x^{2} \cdot 4 - 3 \cdot 4)

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

(x^{3 + 2} + 2 x^{2 + 2} - 3 \cdot x^{2}) + (x^{3} \cdot 4 + 2 x^{2} \cdot 4 - 3 \cdot 4)

AddTerms

Add terms

(x^{5} + 2 x^{4} - 3 \cdot x^{2}) + (x^{3} \cdot 4 + 2 x^{2} \cdot 4 - 3 \cdot 4)

Combine Like Terms and Simplify

Finally, combine like terms and perform all the required operations to simplify the result.

(x^{5} + 2 x^{4} - 3 \cdot x^{2}) + (x^{3} \cdot 4 + 2 x^{2} \cdot 4 - 3 \cdot 4)

Multiply

(x^{5} + 2 x^{4} - 3 x^{2}) + (4 x^{3} + 8 x^{2} - 12)

CommutativePropAdd

Commutative Property of Addition

x^{5} + 2 x^{4} + 4 x^{3} - 3 x^{2} + 8 x^{2} - 12

AssociativePropAdd

Associative Property of Addition

x^{5} + 2 x^{4} + 4 x^{3} + (- 3 x^{2} + 8 x^{2}) - 12

AddTerms

Add terms

x^{5} + 2 x^{4} + 4 x^{3} + 5 x^{2} - 12

Note that multiplying a polynomial with

n

terms by a polynomial with

m

terms produces

n \cdot m

products. When two polynomials are multiplied, the product is a new polynomial whose degree equals the sum of the degrees of the polynomial factors.

$Animation showing the product of two polynomials in standard form. The degree of the product is d_1+d_2. The leading coefficient is (a_{n})(b_{m})x^{d_1+d_2}. There are n times m products.$

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.

Method

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.

P (x) Q (x) = x^{3} + 2 x^{2} - 3 = x^{2} + 4

To multiply these two polynomials, the following four steps can be followed.

Determine the Dimensions of the Table

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
$P (x) = x^{3} + 2 x^{2} - 3$	$3$
$Q (x) = x^{2} + 4$	$2$

For example, a table with $3$ rows and $2$ columns is needed to multiply $P (x)$ by $Q (x) .$

Write the Row and Column Labels of the Table

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.

The terms x^3,2x^2, and -3 written to the left of the table. Also, the terms x^2 and 4 written above the table.

Multiply the Terms to Fill the Table

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

x^{3}

and

x^{2} .

The remaining cells can be filled by following the same procedure.

Performing the required products to fill the table.

Add Terms and Simplify

Finally, add all the expressions inside the table and combine like terms, if any.

Adding all the terms and combining like terms. The result is x^5+2x^4+4x^3+5x^2-12

The product of these polynomials has been found to be

x^{5} + 2 x^{4} + 4 x^{3} + 5 x^{2} - 12 .

Method

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

F irst,

O uter,

I nner,

and

L ast .

Consider, for example, the following product.

(x + 6) (3 x - 2)

These two binomials can be multiplied by following five steps.

Multiply the First Terms

Start by multiplying the first terms of each binomial. In this case, multiply

x

3 x .

(x + 6) (3 x - 2) = x (3 x) - 2 x + 18 x - 12

The empty box is there as a reminder that there are still missing terms.

Multiply the Outer Terms

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

x

- 2 .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) 18 x - 12

Multiply the Inner Terms

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

6

3 x .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) - 12

Multiply the Last Terms

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

6

- 2 .

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) + 6 (- 2)

Simplify

Finally, find each product and combine like terms, if any, to simplify the resulting expression.

(x + 6) (3 x - 2) = x (3 x) + x (- 2) + 6 (3 x) + 6 (- 2)

CommutativePropMult

Commutative Property of Multiplication

(x + 6) (3 x - 2) = 3 x \cdot x + x (- 2) + 6 (3 x) + 6 (- 2)

ProdToPowTwoFac

$a \cdot a = a^{2}$

(x + 6) (3 x - 2) = 3 x^{2} + x (- 2) + 6 (3 x) + 6 (- 2)

MultPosNeg

$a (- b) = - a \cdot b$

(x + 6) (3 x - 2) = 3 x^{2} - x (2) + 6 (3 x) - 6 (2)

Multiply

(x + 6) (3 x - 2) = 3 x^{2} - 2 x + 18 x - 12

AddTerms

Add terms

(x + 6) (3 x - 2) = 3 x^{2} + 16 x - 12

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.

Rule

Closure Property of Polynomial Multiplication

Given two polynomials $P (x)$ and $Q (x),$ the product $P (x) \cdot Q (x)$ 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

P (x)

and

Q (x)

written in standard form.

P (x) Q (x) = a_{n} x^{d_{1}} + \dots + a_{1} x + a_{0} = b_{n} x^{d_{2}} + \dots + b_{1} x + b_{0}

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

P (x)

and

Q (x)

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.

P (x) Q (x) = c_{k} x^{D} + \dots + c_{1} x + c_{0}

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.

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Proof