A polynomial is an expression of the form
a_{0}x^{n} + a_{1}x^{n1} + a_{2}x^{n2} + … + an_{2}x^{2 }+^{ }a_{n1}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 nth 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
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 = (73) 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 + (83)y + (7+12) = 5 y3 – 8 y2 +5y + 19 Difference : (y3 5y2 +8y + 7)  (4y3  3y2 3y + 12) = = (14) y3 + (5 +3) y2 + (8+3)y + (712) = 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
