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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.

A polynomial is an algebraic expression that is the sum of multiple monomials or The largest exponent indicates the polynomial's degree, which in this case is 5. Polynomials with two or three terms are called binomials and trinomials, respectively. When there are more than three terms, the name

terms.Consider the following example polynomial, written in standard form.

polynomialis used more generally.

Calculate the difference between the polynomials.

$3x_{2}−4andx_{2}−4x+2$

Show Solution

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.
Thus, the difference is 2x2+4x−6.

$(3x_{2}−4)−(x_{2}−4x+2)$

Distr

Distribute -1

3x2−4−x2+4x−2

CommutativePropAdd

Commutative Property of Addition

3x2−x2+4x−4−2

SimpTerms

Simplify terms

2x2+4x−6

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

### 2

### 3

### 4

Finally, combine like terms and perform all the required operations to simplify the result.
Note that multiplying a polynomial with $n$ terms by a polynomial with $m$ terms produces $n⋅m$ 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.

Distribute One Polynomial to All the Terms of the Other

Start by writing the product P(x)⋅Q(x).
Next, distribute P(x) to each term of Q(x).

Clear Parenthesis by Applying the Distributive Property

Apply the Distributive Property one more time to clear all the parentheses.

Apply the Product of Powers Property

Using the Product of Powers Property to rewrite some products as one single power.

Combine Like Terms and Simplify

x5+2x4−3⋅x2+x3⋅4+2x2⋅4−3⋅4

Multiply

Multiply

x5+2x4−3x2+4x3+8x2−12

CommutativePropAdd

Commutative Property of Addition

x5+2x4+4x3−3x2+8x2−12

AssociativePropAdd

Associative Property of Addition

x5+2x4+4x3+(-3x2+8x2)−12

AddTerms

Add terms

x5+2x4+4x3+5x2−12

Two polynomials can also be multiplied using the Box Method and the FOIL Method, however, the latter is useful only for multiplying binomials. Keep in mind that these two methods are based on the Distributive Property.

Find a polynomial that represents the area of the rectangle.

Show Solution

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.
Thus, the area can be represented by the polynomial

$A=ℓw$

SubstituteExpressions

Substitute expressions

A=(2x+2)(3x−5)

MultPar

Multiply parentheses

A=6x2−10x+6x−10

SimpTerms

Simplify terms

A=6x2−4x−10

A=6x2−4x−10.

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