# Polynomials - Definitions, Adding and Substracting Polynomials

A polynomial is an expression of the form

*a*_{0}*x*^{n}* + a*_{1}*x*^{n-1}* + a*_{2}*x*^{n-2}* + … + an-*_{2}*x*^{2 }*+*^{ }*a*_{n-1}*x + a*_{n}_{.}

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*a*_{0}*, a*_{1}* … a*_{n} in expression P1, or *4, 7, -11, 2 *above* *are constants and are called coeficients

x is variable; note that a polynomial can have multiple variables, for example *5x*^{2}*y*^{3}* - 5xy +7 *is valid polynomial

The polynomial expression consists of terms as shown. Polynomial terms have variables which are raised to whole positive number exponents or zero (if raised to a zero exponent, the terms are just constants like *-7* above since *-7.x*^{0 }*= -7 . 1 = -7*); in the terms there are no square roots of variables or other fractional powers (ftactional power like *x*^{1/n} is the n-th root of x and not allowed in polynomials), and no variables in the denominator of any fractions (which is same as no negative powers as *x*^{-5}* = 1/(x*^{5}*)* ). Here are some examples:

A polynomial of one term is called a monomial. A polynomial of two terms is

called a binomial. A polynomial of three terms is called a trinomial.

**example: ***7, 8x*^{2}*, - x*^{7}*y*^{3} are mononomials

*2x+8, 5x*^{4}* +7x*^{3} are binomials

*3x*^{3}*y*^{4}* + 7x*^{2}*y +11, x*^{7}*+4x*^{3}*+5* are trinomials

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The **degree of a term** is the exponent of the variable in the term, or in case of multiple variables the degree of the term is the sum of the exponents of the variables.

The **degree of a polynomial** is the largest of the degrees of the individual terms.

**example: **The term *5x*^{9} has degree *9*

The term *12xy*^{4}*z*^{5} has degree *10 *(the sum of exponents of* x, y, z - 1+4+5)*

The polynomial *5 + x*^{7}* + x *has degree *7*

**Like and unlike terms.** Two terms are called like terms if they have the same variables raised to the same exponents. Like terms may differ only in the coefficient of the terms. Terms that are not like terms are called unlike terms.

**example: ***2x* and *2* – unlike terms

*5x* and *3y* - unlike terms

*8x*^{2} and *8x*^{3} - unlike terms

*5x*^{7} and *11x*^{7} – like terms

*3x*^{2}*y*^{3} and *-10 x*^{2}*y*^{3} – like terms

When **simplifying** a polynomial **only like terms** can be combined as shown:

*3x*^{2}*y*^{3 }*- 10 x*^{2}*y*^{3 }*= (3 – 10) x*^{2}*y*^{3}* = -7 x*^{2}*y*^{3}

Note that allows us to combine like terms is the fact that the variable portion of like terms is a common factor, allowing us to apply the reverse distributive law: *ac + bc = (a + b)c*

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**example: **Simplify *7x*^{3}* + 3x*^{2}* – 4 – 3x*^{3}* + 8x*^{2}* + 9x +2*

*7x*^{3}* + 3x*^{2}* – 4 – 3x*^{3}* + 8x*^{2}* + 9x +2 =*

*= (7x*^{3}*– 3x*^{3}*) + (3x*^{2}*+ 8x*^{2}*) + (9x) + (-4 + 2) *Group like terms

*= (7-3) x*^{3}* + (3+8) x*^{2}* + (9)x + (-4+2)* Apply reverse distributive law

*= 4 x*^{3}* + 11 x*^{2}* + 9x - 2*

**Note that only like terms can be combined**

**Addition and Substraction of polynomials**

The sum or difference (the result of addition and substraction) of polynomials is found by combining like terms similarly to the example of simplifying above.

**example: **Find the sum and difference of

polynomial1: *y3 -5y2 +8y + 7* and polynomial2: *4y3 - 3y2 - 3y + 12*

Sum:* (y3 -5y2 +8y + 7) + (4y3 - 3y2 -3y + 12) =*

*= (1+4) y3 + (-5 -3) y2 + (8-3)y + (7+12)*

*= 5 y3 – 8 y2 +5y + 19*

Difference :* (y3 -5y2 +8y + 7) - (4y3 - 3y2 -3y + 12) =*

*= (1-4) y3 + (-5 +3) y2 + (8+3)y + (7-12)*

*= -3 y3 – 2 y2 + 5y – 5*

Note that *y3 + 4y3 = 1.y3 + 4.y3 = (1+4) y3*^{ }; coefficient of 1 is not written in most cases, but has to be applied to the sum or difference when grouping

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