8­1 Adding and Subtracting Polynomials poly​
­(prefix) p a h l ee Related Words:​
many, several, polynomial, polygon Definition:​
The prefix poly means “many” or “several” and is used to make compound words such as polygon. Main Idea:​
Polynomials may contain one or several monomials. A ​
Monomial​
is a real number, a variable, or a product of a real number and one or more variables with whole­number exponents. Here are some examples of monomials. 18
− 4x2
z a
3 2.5xy3
Essential Understanding​
You can use monomials to form larger expressions called ​
polynomials​
. polynomials can be added and subtracted. The ​
degree of a monomial​
is the sum of the exponents of its variables. The degree of a nonzero constant is 0. Zero has no degree. Find the degree of each monomial. A.) 8a3 B.) 2b8 c2 C.) − 3 D.) 0 You can add or subtract monomials by adding or subtracting like terms. Simplify E.) 2m3 n3 + 9m3 n3 F.) 3t4 + 11t4 G.) 30v4 w3 − 12v4 w3 H.) 5bc4 − 13bc4 A ​
polynomial​
is a monomial or a sum of monomials. The following polynomial is the sum of the monomials 3x4 , 5x2 , − 7x , and 1 . The polynomial shown above is in ​
standard form​
. ​
Standard form of a polynomial ​
means that the degrees of its monomial terms decrease from left to right. The ​
degree of a polynomial​
in one variable is the same as the degree of the monomial with the greatest exponent. The degree of 3x4 + 5x2 − 7x + 1 is 4.
Write each polynomial in standard form. Then name each polynomial based on its degree and number of terms. I.) − 2q + 7 J.) − 7x2 − 4x + 4 K.) 3z4 − 5z − 2z2 You can add polynomials by adding like terms. Simplify. L.) 6x2 + 7 (
)
M.) (5x2 + 3) + 15x2 + 2 + 3x2 + 1 Education​
The number of students at East High School and the number of students at Central High School over a 10 year period can be modeled by the following polynomials. East High School: − 11x2 + 133x + 1200 Central High School: − 7x2 + 95x + 1100 In each polynomial, x = 0 corresponds to the first year in the 10­year period. What polynomial models the total number of students at both high schools? Recall that subtraction means to add the opposite. So when you subtract a polynomial, change each of the terms to its opposite. Then add the coefficients. N.) 6x3 + 17 ​
­ 4x3 + 9 (
) (
)
O.) 14h4 + 3h3 − 9h4 + 2h3