Algebraic Expressions; Polynomials

1. Algebraic Expressions; Polynomials • A variable is a symbol to which we can assign values. • An algebraic expression is a string of symbols which includes variables, constants, and algebraic operations (such as +, , , ). • An algebraic expression takes on a value when we assign a specific number to each variable in the expression. Thus, the expression is evaluated when, say, m = 3and n = 2 are substituted into the expression. We obtain

2. Powers • An expression in which a variable multiplies itself repeatedly is often used. • In the expression where n is a natural number and a is a real number is called the nth power of a. We call a the base and n the exponent. • Rule for multiplying powers: n factors

3. Polynomials in x • Let x denote a variable and let n be a constant, nonnegative integer. The expression axn, where a is a constant real number, is called a monomial in x. • A polynomial in x is an expression that is a sum of monomials and has the general form: • Each of the monomials in the previous equation is called a term of P, and the numbers are called the coefficients of P. • The degree of P is the largest exponent, which is n. The zeropolynomial has all coefficients equal to 0, and it has no degree.

4. Polynomials in x and y • Let x and y denote variables and let m and n be constant, nonnegative integers. The expression axmyn, where a is a constant real number, is called a monomial in x and y. • A polynomial in x and y is an expression that is a sum of monomials in x and y. • The degree of the monomial axmyn is m + n. The degree of a polynomial in x and y is the degree of its highest degree nonzero monomial. • What is the degree of the following polynomial in x and y ?

5. Operations with Polynomials in x • Like terms of polynomials have the same exponent of x. • Two polynomials are equal if all the like terms are equal. • Example. x + x2 = x2 + x. • If P and Q are polynomials in x, the sumP + Q is obtained by forming the sum of all pairs of like terms. Similarly, the differenceP – Q is obtained by forming the differences of like terms. • Problem. If P = x2 + x and Q = 3x2– x, find P + Q and P – Q. • Note--similar definitions apply to polynomials in x and y.

6. Multiplication of Polynomials • Every term in one polynomial P is multiplied times every term in the second polynomial Q and then like terms are added to form the product PQ. • Example. • Problem. Compute the product:

7. Special Products of Polynomials in aand b • Be sure to memorize the following: • Problem. Multiply Solution. By the first equation given above,

8. Summary of Algebraic Expressions, Polynomials; We discussed • Variables, algebraic expressions, and evaluation of algebraic expressions • The nth power of an expression, its base and its exponent • The rule for multiplying powers • Polynomials in x and polynomials in x, y • The degree of a polynomial • The sum, difference, and product of two polynomials • A number of special products including the product of the sum and difference of two numbers and the square of a binomial