menu_book {{ printedBook.name}}

arrow_left {{ state.menu.current.label }}

arrow_left {{ state.menu.current.current.label }}

arrow_left {{ state.menu.current.current.current.label }}

{{ result.displayTitle }} *navigate_next*

{{ result.subject.displayTitle }}

{{ 'math-wiki-no-results' | message }}

{{ 'math-wiki-keyword-three-characters' | message }}

{{ r.avatar.letter }}

{{ r.name }} {{ r.lastMessage.message.replace('TTREPLYTT','') }} *people*{{keys(r.currentState.members).length}} *schedule*{{r.lastMessage.eventTime}}

{{ r.getUnreadNotificationCount('total') }} +

{{ u.avatar.letter }}

{{ u.displayName }} (you) {{ r.lastMessage.message.replace('TTREPLYTT','') }} *people*{{keys(r.currentState.members).length}} *schedule*{{r.lastMessage.eventTime}}

{{ r.getUnreadNotificationCount('total') }} +

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 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. ### Classifying Polynomials by Their Number of Terms

When there are more than three terms, the name

$Example Polynomial✓5xy+2xy_{2}+8x−9Not a Polynomial×3xy+2xy_{-2}+7xy_{3}−1 $

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

polynomialis commonly used.

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

$-2x_{2}+x+3and2x_{2}+4x−10 $

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

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.
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

CommutativePropAdd

Commutative Property of Addition

(-2x2+2x2)+(x+4x)+(3−10)

(-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

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

3

Simplify the Result

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.
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.

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.

$+ -2x_{2}+x+3-2x_{2}+4x−10-0x_{2}+5x−7( $

By comparing the results, it can be concluded that no matter which format of addition is used, the sums are the same.
$3x_{2}−4andx_{2}−4x+2$

Show Solution *expand_more*

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

- 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.

Find the product of the polynomials x3+x2+5 and x4+2x+1 by distributing (x4+2x+1) to each term in x3+x2+5. Then, determine the degree of the resulting polynomial.
It can be seen that since both polynomials have 3 terms, multiplying them results in 3×3=9 products. Nevertheless, it can be simplified by combining like terms.
It can also be noted that the polynomial $x_{3}+x_{2}+5$ its of degree 3 and the polynomial $x_{4}+2x+1$ its of degree 4. When written in standard form, their product is
This is a polynomial of degree 3+4=7.

Find a polynomial that represents the area of the rectangle.

Show Solution *expand_more*

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.

{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

{{ focusmode.exercise.exerciseName }}

close

Community rate_review

{{ r.avatar.letter }}

{{ u.avatar.letter }}

+