An algebraic expression is one that contains one or more variables. To evaluate an algebraic expression means to replace the variable of variables in the expression with given numbered values and then simplify the resulting numerical expression.
On type of algebraic expression, in which the terms are all monomials, is called a polynomial. The four fundamental operations can be performed using essentially the same ideas. The commutative, associative, and distributive properties provide the basis for rearranging, regrouping, and combining similar terms in the addition and subtraction of polynomial expressions. To add polynomials, simply combine like terms. To subtract polynomials, simply add the opposite of the second polynomial to the first and combine liker terms. The final resulting polynomial in each case should be written in decreasing order according to the power of the variable. Either a vertical or horizontal format can be used for either operation. For example we can combine each of the following expressions using the indicated operations in vertical or horizontal format.
Multiplying polynomials involves the use of the properties of exponents and the distributive property. In general to find the product of two polynomials, multiply each term of the first polynomial by each term of the second polynomial, combine similar terms and write the resulting polynomial in decreasing order. If there are more than two polynomials to be multiplied, multiply the product of the first two by the next polynomial, combine like terms, and write the final product in decreasing order.
To divide a polynomial by a monomial, divide each term in the numerator by the denominator and write the sum of the quotients. To divide a polynomial by a polynomial (degree 2 or more) use a method similar to that used for division of whole numbers. In particular, first arrange the terms in the divisor and dividend in decreasing powers of the variable, filling in any missing power of the variables in the dividend with +0 times the missing power of the variable. Then, the next step is to divide the first term of the divisor into the first term of the dividend. Multiply the quotient from the division by each term in the divisor and subtract the products of each term from the dividend. The result (difference) is a new dividend. Repeat the last step using the divisor and the new dividend again until we obtain a remainder which is of degree less than that of the divisor or zero. The final quotient is the sum of the quotients obtained from each step plus the remainder expressed as a fraction