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Various events in daily life depend on multiple factors. Consider the $CO_{2}$ emission of a car. It depends on many variables, such as the gas used, the engine size, and the number of cylinders. Countless daily events like that can be modeled using a combination of powers of different variables. Making those models requires using algebraic expressions with multiple terms called *polynomials*.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Take a look at the following single-term algebraic expressions. Can a shared characteristic be recognized which would allow for them to be classified reasonably? Drag the expressions and use the colored areas to visualize any common themes, and identify which expressions could be grouped together.

Discussion

A monomial is an algebraic expression consisting of only one term. It is a product of powers of variables and a constant called the coefficient.

A single-term expression is a monomial only if all of its variables have whole numbers — non-negative and integers — as exponents. However, variables with positive exponents in the denominator are excluded because they are equivalent to a power in the numerator with the opposite exponent, according to the Quotient of Powers Property. Consider the following example.$y_{2}5x =5xy_{-2} $

In other words, if a variable in the denominator of an expression, it is not a monomial. The following are valid examples of monomials. Expression | Why It Is a Monomial |
---|---|

$5$ | Any constant is a valid monomial. By the Zero Exponent Property, $5x_{0}=5.$ |

$0$ | The coefficient of a monomial can be $0.$ |

$-2x_{5}$ | The coefficient can be negative. |

$5x_{3}y $ | A monomial can have numbers in the denominator. |

Although they appear to be monomials at first glance, the single-term expressions in the following table do not satisfy the definition of a monomial.

Expression | Why It Is Not a Monomial |
---|---|

$2x_{-1}$ | The variables of a monomial cannot have negative integer exponents. |

$4yx_{3} $ | Monomials cannot have variables in the denominator. |

$5x_{3}y_{21}$ | The variables of a monomial must only have whole number exponents. |

Concept

The degree of a monomial is the sum of the exponents of its variable factors. If a variable has no exponent written in it, it is assumed to be $1.$ Additionally, all nonzero constants have a degree of $0.$ The constant $0$ does not have a degree.

Monomial | Degree |
---|---|

$3x$ | $1$ |

$x_{2}$ | $2$ |

$9x_{3}$ | $3$ |

$x_{3}y$ | $4$ |

$7$ | $0$ |

$13a_{3}b_{4}c_{5} $ | $12$ |

$0$ | undefined |

Pop Quiz

The next applet shows different algebraic expressions. Use the new knowledge introduced previously to determine whether or not these expressions are monomials.

Pop Quiz

Discussion

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. | $3x_{2}y_{3}$ |

Binomial | A polynomial with a exactly two terms. | $5xy+3x_{2}y_{3}$ |

Trinomial | A polynomial with exactly three terms. | $x_{3}−9x+4$ |

polynomialis commonly used.

Concept

The degree of a polynomial is defined as the greatest of the degrees of its terms. For example, the following is a polynomial of degree $5.$
*zero polynomial*, the degree is undefined. The following table summarizes this information by providing some specific examples.

Depending on the degree of a polynomial, it can have different characteristics. For example, a polynomial with degree $1$ is linear, and a polynomial with degree $2$ is quadratic.

The monomial with the highest degree is $4xy_{4}.$ This means that the degree of the polynomial is $1+4=5.$ In other cases, if a polynomial consists of only a nonzero constant term, the polynomial has degree $0.$ For the special case of the

Polynomial | Degree |
---|---|

$-5x_{2}y+x_{2}y_{4}−11x_{2}−3$ | $6$ |

$x−11x_{4}+8x_{3}$ | $4$ |

$7$ | $0$ |

$0$ | Undefined |

Concept

In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Pop Quiz

The following applet alternates between different algebraic expressions. Determine whether or not these expressions are polynomials.

Pop Quiz

The following applet alternates between different polynomials. Determine the given polynomial's degree or leading coefficient as indicated.

Example

Izabella is improving her understanding of polynomials in her algebra course. However, she has come across some problems where she needs to identify the number of terms of some algebraic expressions and only some of the expressions are polynomials.

Izabella feels confused. Help her solve these problems.{"type":"pair","form":{"alts":[[{"id":5,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.20188em;vertical-align:-0.8804400000000001em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><span class=\"mord mathdefault\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.008548em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.008548em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">-<\/span><\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.9540200000000001em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.7935600000000003em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">7<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span>"},{"id":4,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.7935599999999998em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">8<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\">x<\/span><\/span><\/span><\/span>"}],[{"id":0,"text":"Expression with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span> terms"},{"id":1,"text":"Expression with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">3<\/span><\/span><\/span><\/span> terms"},{"id":5,"text":"Expression with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span> terms"},{"id":2,"text":"Polynomial with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span> terms"},{"id":3,"text":"Polynomial with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">3<\/span><\/span><\/span><\/span> terms"},{"id":4,"text":"Polynomial with <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span> terms"}]],"lockLeft":false,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[5,3,1,2,0,4],[0,1,5,2,3,4]]}

For an algebraic expression to be a polynomial, each term must be a valid monomial. A monomial is a single-term expression that is a product of numbers and variables raised to exponents that are whole numbers. Three of the given expressions do not satisfy this definition. That takes them out of the running as being polynomials.

Expression | Reason |
---|---|

$4x+x_{21}$ | It has a non-integer exponent. |

$2x_{2}+5+y_{-2}$ | It has a negative exponent. |

$2y3x +x_{2}−5y_{3}+9$ | It has a variable in the denominator. |

Now that Izabella knows which expressions are polynomials and which are not, she can figure out how to identify their number of terms. Recall that, in any algebraic expression, terms are separated by a plus or a minus sign.

Polynomial | Number of terms |
---|---|

$4x −3x_{3}$ | $2$ |

$2x_{2}+5−7y$ | $3$ |

$8xy +3x_{3}−3y_{3}+2x$ | $4$ |

Expression | Number of terms |

$4x+x_{21}$ | $2$ |

$2x_{2}+5+y_{-2}$ | $3$ |

$2y3x +x_{2}−5y_{3}+9$ | $4$ |

Discussion

Concept

A polynomial in one variable is expressed in standard form when the monomials that form it are arranged in decreasing degree order. This form can be represented with the following general expression.

$a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{1}x_{1}+a_{0}x_{0}$

$x_{5}−12x_{4}−2x_{3}+8x_{2}+9x+0 $

It should be noted that coefficients can be zero. In those cases, the corresponding terms are often omitted, which causes consecutive terms to have exponents that are $3x_{4}−2x_{2}+6x+5⇕3x_{4}+0⋅x_{3}−2x_{2}+6x+5 $

It can be seen that the $x_{3}-$term was omitted, at first. However, if chosen to do so, the polynomial can be expressed in standard form with a coefficient of $0$ and the corresponding term.Concept

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s).

$7xy+2x−3xy−2x_{2}+x+5x_{2} $

In this expression, there are three sets of like terms — namely, $xy-$terms, $x-$terms, and $x_{2}-$terms.
Note that all constants are like terms. Like terms in an expression can be combined by using addition or subtraction to rewrite the expression in simplest form.Method

In an expression, groups of like terms can be combined to write an equivalent expression with the least possible number of terms. As an example, consider the following expression.
*expand_more*
*expand_more*
*expand_more*

$7xy+2x−3xy−2x_{2}+x+5x_{2} $

There are three steps to follow to combine like terms.
1

Identify Like Terms

2

Grouping Like Terms

Now that the like terms are identified, use the Commutative Property of Addition to rearrange the expression so that the like terms are grouped together.

3

Add and Subtract the Coefficients

Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.

Now that all like terms have been combined, the initial expression has been reduced to its simplest form.

$7xy+2x−3xy−2x_{2}+x+5x_{2}⇕4xy+3x+3x_{2} $

Example

Izabella, so excited about solving polynomial related problems, has started a study group with Magdalena to work together on another set of polynomials. This time, the polynomials need to be written in standard form and have their characteristics identified.

Once again, they have run into a problem. The polynomials are not simplified yet. Join their study group and give them a hand.$ I.x_{2}+4x_{3}−5x_{3}+6x_{2}+2x_{3}II.8x+4x_{3}−2x_{5}+7x−12x_{3} $

a Simplify Polynomial **I,** then write it in standard form. What type of polynomial is this, according to its number of terms? Lastly, find its degree and leading coefficient.

b Simplify Polynomial **II,** then write it in standard form. What type of polynomial is this according to its number of terms? Lastly, find its degree and leading coefficient.

a **Standard Form: ** $x_{3}+7x_{2}$

**Leading Coefficient: ** $1$

**Classification: ** Binomial

**Degree: ** $3$

b **Standard Form: ** $-2x_{5}−8x_{3}+15x$

**Leading Coefficient: ** $-2$

**Classification: ** Trinomial

**Degree: ** $5$

a To simplify the polynomial, like terms will be identified and grouped.

$x_{2}+4x_{3}−5x_{3}+6x_{2}+2x_{3}⇕x_{2}+6x_{2}+4x_{3}−5x_{3}+2x_{3} $

Now, like terms will be combined by adding or subtracting their coefficients.
$x_{2}+6x_{2}+4x_{3}−5x_{3}+2x_{3}$

FactorOut

Factor out $x_{2}&x_{3}$

$(1+6)x_{2}+(4−5+2)x_{3}$

AddSubTerms

Add and subtract terms

$7x_{2}+x_{3}$

Term | Degree | Coefficient |
---|---|---|

$7x_{2}$ | $2$ | $7$ |

$x_{3}⇔(1)x_{3}$ | $3$ | $1$ |

$Standard Formx_{3}+7x_{2} $

Finally, recall that the degree and leading coefficient of a polynomial are the same as those of the monomial of highest degree. Since the highest degree monomial is $(1)x_{3},$ the polynomial's degree is $3$ and its leading coefficient is $1.$ The following table summarizes all the information found. Standard Form | ||
---|---|---|

$x_{3}+7x_{2}$ | ||

Classification | Degree | Leading Coefficient |

Binomial | $3$ | $1$ |

b The same steps as in Part A will be followed. The polynomial will be simplified by identifying and combining like terms.

$8x+4x_{3}−2x_{5}+7x−12x_{3}$

CommutativePropAdd

Commutative Property of Addition

$8x+7x+4x_{3}−12x_{3}−2x_{5}$

FactorOut

Factor out $x&x_{3}$

$(8+7)x+(4−12)x_{3}−2x_{5}$

AddSubTerms

Add and subtract terms

$15x−8x_{3}−2x_{5}$

Term | Degree | Coefficient |
---|---|---|

$15x⇔15x_{1}$ | $1$ | $15$ |

$-8x_{3}$ | $3$ | $-8$ |

$-2x_{5}$ | $5$ | $-2$ |

$Standard Form-2x_{5}−8x_{3}+15x $

Finally, since the highest degree monomial is $-2x_{5},$ the polynomial's degree is $5$ and its leading coefficient is $-2.$ The information found will be summarized using a table. Standard Form | ||
---|---|---|

$-2x_{5}−8x_{3}+15x$ | ||

Classification | Degree | Leading Coefficient |

Trinomial | $5$ | $-2$ |

Closure

Now that polynomials have been introduced, equations containing them can be managed. The following example shows a *polynomial equation,* also known as an *algebraic equation*. *Fundamental Theorem of Algebra*. Since the example polynomial equation above is of degree $3,$ the associated equation has $3$ solutions.

Polynomial equations can be solved algebraically by using methods such as the Quadratic Formula or others based on the Zero Product Property. Alternatively, they can also be solved graphically or by using numerical methods.

$2x_{3}−12=-8x_{2}−2x $

It is always possible to rearrange these types of equations so that the right-hand side is $0.$ $2x_{3}+8x_{2}+2x−12=0 $

In this format, the degree of the polynomial in the left-hand side defines the degree of the equation. It is a remarkable fact that the number of solutions of a polynomial equation is equal to its degree. This relationships is known as the Polynomial Equation | Degree |
---|---|

$2x_{3}+8x_{2}+2x−12=0$ | $3$ |

Solutions | |

$x=-3,x=-2,x=1$ |

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