Polynomials - Definitions, Adding and Substracting Polynomials
A polynomial is an expression of the form
a0xn + a1xn-1 + a2xn-2 + … + an-2x2 + an-1x + an.
a0, a1 … an 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 5x2y3 - 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.x0 = -7 . 1 = -7); in the terms there are no square roots of variables or other fractional powers (ftactional power like x1/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/(x5) ). 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, 8x2, - x7y3 are mononomials
2x+8, 5x4 +7x3 are binomials
3x3y4 + 7x2y +11, x7+4x3+5 are trinomials
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 5x9 has degree 9
The term 12xy4z5 has degree 10 (the sum of exponents of x, y, z - 1+4+5)
The polynomial 5 + x7 + 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
8x2 and 8x3 - unlike terms
5x7 and 11x7 – like terms
3x2y3 and -10 x2y3 – like terms
When simplifying a polynomial only like terms can be combined as shown:
3x2y3 - 10 x2y3 = (3 – 10) x2y3 = -7 x2y3
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 7x3 + 3x2 – 4 – 3x3 + 8x2 + 9x +2
7x3 + 3x2 – 4 – 3x3 + 8x2 + 9x +2 =
= (7x3– 3x3) + (3x2+ 8x2) + (9x) + (-4 + 2) Group like terms
= (7-3) x3 + (3+8) x2 + (9)x + (-4+2) Apply reverse distributive law
= 4 x3 + 11 x2 + 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