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# Manipulating 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.

## Polynomial

A polynomial is an algebraic expression that is the sum of multiple monomials or terms. Consider the following example polynomial, written in standard form.
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 polynomial is used more generally.
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Exercise
Calculate the difference between the polynomials.
Show Solution
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.
3x24x2+4x2
3x2x2+4x42
2x2+4x6
Thus, the difference is 2x2+4x6.

## 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
Start by writing the product P(x)Q(x).
Next, distribute P(x) to each term of Q(x).
(x3+2x23)(x2+4)
(x3+2x23)x2+(x3+2x23)4

### 2

Clear Parenthesis by Applying the Distributive Property
Apply the Distributive Property one more time to clear all the parentheses.
(x3+2x23)x2+(x3+2x23)4
x3x2+2x2x23x2+x34+2x2434

### 3

Apply the Product of Powers Property
Using the Product of Powers Property to rewrite some products as one single power.
x3x2+2x2x23x2+x34+2x2434
x5+2x43x2+x34+2x2434

### 4

Combine Like Terms and Simplify
Finally, combine like terms and perform all the required operations to simplify the result.
x5+2x43x2+x34+2x2434
x5+2x43x2+4x3+8x212
x5+2x4+4x33x2+8x212
x5+2x4+4x3+(-3x2+8x2)12
x5+2x4+4x3+5x212
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 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.
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Exercise

Find a polynomial that represents the area of the rectangle.

Show Solution
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, 3x5. We find the polynomial representing the area by multiplying the length and the width.
A=(2x+2)(3x5)
A=6x210x+6x10
A=6x24x10
Thus, the area can be represented by the polynomial
A=6x24x10.