Adding Subtracting and Multiplying Polynomials

The word polynomial is used to describe a specific type of expression. Although polynomials contain multiple terms, they behave similar to numbers and can be manipulated as such.

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Concept

Polynomial

A polynomial is either a single monomial or a sum of them. Each of the monomials that form the polynomial are referred to as a term of the polynomial. This definition implies that, for an expression to be a polynomial, each of its terms must be a valid monomial.

$$\begin{array}{c}\text{Example Polynomial }✓\\ 5xy+2x{y}^2+8x-9\\ \\ \text{Not a Polynomial }\times \\ 3xy+2x{y}^{\text{-}2}+7x{y}^3-1\end{array}$$

In the above example, the latter expression is not a polynomial because the term $2x{y}^{\text{-}2}$ is not a monomial. The variables in a monomial can only have whole numbers as exponents.

Classifying Polynomials by Their Number of Terms

One way that polynomials can be classified is according to the number of terms they have. The following table shows the names used for this classification.

Name	Definition	Example
Monomial	A polynomial with a single term.	3x2y3
Binomial	A polynomial with a exactly two terms.	5xy+3x2y3
Trinomial	A polynomial with exactly three terms.	x3−9x+4

When there are more than three terms, the name polynomial is commonly used.

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Method

Adding and Subtracting Polynomials

Given two polynomials, their sum or difference can be calculated. Consider the following pair of polynomials.

$$\begin{array}{c}\text{-}2x^2+x+3\text{and}2x^2+4x-10\end{array}$$

In order to add these polynomials, there are three steps to follow. Note that subtraction of the polynomials can be performed by applying the same three steps, only instead of adding the like terms, they will be subtracted.

1

Rearrange Terms

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Generally speaking, polynomials are added by adding their like terms. Therefore, it can be helpful to rearrange the terms so that the like terms are next to each other.

(-2x2+x+3)+(2x2+4x−10)

RemovePar

Remove parentheses

-2x2+x+3+2x2+4x−10

CommutativePropAdd

Commutative Property of Addition

(-2x2+2x2)+(x+4x)+(3−10)

When subtracting polynomials, remember to distribute -1 to the terms of the second polynomial before rearranging the terms.

(-2x2+x+3)−(2x2+4x−10)

RemovePar

Remove parentheses

-2x2+x+3−(2x2+4x−10)

Distr

Distribute -1

-2x2+x+3−2x2−4x+10

CommutativePropAdd

Commutative Property of Addition

(-2x2−2x2)+(x−4x)+(3+10)

2

Add and Subtract Like Terms

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Next, the like terms can be combined. In other words, the monomials should be added or subtracted by adding or subtracting their coefficients.

(-2x2+2x2)+(x+4x)+(3−10)

AddSubTerms

Add and subtract terms

0x2+5x+(-7)

3

Simplify the Result

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Finally, the obtained result of addition can be simplified. For example, since the coefficient before x2-term is 0, this term is equal to 0 and can be omitted.

0x2+5x+(-7)

ZeroPropMult

Zero Property of Multiplication

5x+(-7)

AddNeg

a+(-b)=a−b

5x−7

Therefore, the sum is 5x−7. Note that the given polynomials are of degree 2. However, the new polynomial is of degree 1 because the x2-terms canceled each other out. If they had not canceled each other out, the new polynomial would have also been of degree 2.

When adding or subtracting two polynomials of degrees m and n, the degree of the resulting polynomial is at most the equal to the degree of higher degree polynomial.

Extra

Vertical Format of Addition and Subtraction

Previously, polynomials were added by using the horizontal format. The vertical format of polynomial addition or subtraction will now be presented. Note that for this method, the like terms should be aligned vertically.

$$\begin{array}{rl}& \text{-}2x^2+x+3\\ +& 2x^2+4x-10\\ & 0x^2+5x-7\end{array}$$

By comparing the results, it can be concluded that no matter which format of addition is used, the sums are the same.

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Example

Subtract the polynomials

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Calculate the difference between the polynomials.

$$3x^2-4\text{and}{x}^2-4x+2$$

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Subtracting polynomials is done by combining like terms. When we calculate the difference, it is important to distribute the negative sign to all terms in the second polynomial.

$\left(3x^2-4\right)-\left(x^2-4x+2\right)$

Distr

Distribute -1

3x2−4−x2+4x−2

CommutativePropAdd

Commutative Property of Addition

3x2−x2+4x−4−2

SimpTerms

Simplify terms

2x2+4x−6

Thus, the difference is 2x2+4x−6.

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Polynomials are multiplied by using the Distributive Property. For example, for the polynomial product it is possible to distribute (c+d+e) to every term of (a+b). Then, by using the Distributive Property once more, the product can be expressed explicitly. When multiplying two polynomials, each term of the first polynomial multiplies with each term of the second one. This have some important consequences:

• The product of a polynomial with n terms and a polynomial with m terms results in n×m products. For example, the product of two binomials gives 2×2=4 products as result.
• Since each product of terms results in a new monomial, the result of a polynomial multiplication is a sum of monomials, which is, by definition, another polynomial. Therefore, polynomials are closed under multiplication.
• When two polynomials are multiplied, the product is a new polynomial with a degree that is the sum of the degrees of the factor polynomials. This follows from the Product of Powers Property when multiplying the highest degree monomials of the factor polynomials.

In order to make the multiplication process systematic there are different methods that can be used, such as the FOIL Method and using tables of products. Ultimately, all of these methods are based on the Distributive Property. An example of a table of products is shown below.

Example

Multiply the polynomials

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Find a polynomial that represents the area of the rectangle.

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The area of a rectangle is calculated using the relationship

Here, the length is represented by the binomial 2x+2 and the width by another binomial, 3x−5. We find the polynomial representing the area by multiplying the length and the width.

$A=\ell w$

SubstituteExpressions

Substitute expressions

A=(2x+2)(3x−5)

MultPar

Multiply parentheses

A=6x2−10x+6x−10

SimpTerms

Simplify terms

A=6x2−4x−10

Thus, the area can be represented by the polynomial

A=6x2−4x−10.

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