


Enter an expression and click the Simplify button. Simplifying PolynomialsIn section 3 of chapter 1 there are several very important definitions, which we have used many times. Since these definitions take on new importance in this chapter, we will repeat them. When an algebraic expression is composed of parts connected by + or  signs, these parts, along with their signs, are called the terms of the expression. a + b has two terms.
When an algebraic expression is composed of parts to be multiplied, these parts are called the factors of the expression. ab has factors, a and b. It is very important to be able to distinguish between terms and factors. Rules that apply to terms will not, in general, apply to factors. When naming terms or factors, it is necessary to regard the entire expression.
An exponent is a numeral used to indicate how many times a factor is to be used in a product. An exponent is usually written as a smaller (in size) numeral slightly above and to the right of the factor affected by the exponent.
Note the difference between 2x^{3} and (2x)^{3}. From using parentheses as grouping symbols we see that 2x^{3} means 2(x)(x)(x), whereas (2x)^{3} means (2x)(2x)(2x) or 8x^{3}.
In an expression such as 5x^{4} Note that only the base is affected by the exponent.
When we write a literal number such as x, it will be understood that the coefficient is one and the exponent is one. This can be very important in many operations. x means 1x^{1}.
MULTIPLICATION LAW OF EXPONENTSOBJECTIVESUpon completing this section you should be able to correctly apply the first law of exponents. Now that we have reviewed these definitions we wish to establish the very important laws of exponents. These laws are derived directly from the definitions. First Law of Exponents If a and b are positive integers and x is a real number, then
For any rule, law, or formula we must always be very careful to meet the conditions required before attempting to apply it. Note in the above law that the base is the same in both factors. This law applies only when this condition is met.
An exponent of 1 is not usually written. When we write x, the exponent is assumed: x = x1. This fact is necessary to apply the laws of exponents. If an expression contains the product of different bases, we apply the law to those bases that are alike. MULTIPLICATION OF MONOMIALSOBJECTIVESUpon completing this section you should be able to:
A monomial is an algebraic expression in which the literal numbers are related only by the operation of multiplication.
To find the product of two monomials multiply the numerical coefficients and apply the first law of exponents to the literal factors.
MONOMIALS MULTIPLIED BY POLYNOMIALSOBJECTIVESUpon completing this section you should be able to:
A polynomial is the sum or difference of one or more monomials.
Special names are used for some polynomials. If a polynomial has two terms it is called a binomial. If a polynomial has three terms it is called a trinomial. In the process of removing parentheses we have already noted that all terms in the parentheses are affected by the sign or number preceding the parentheses. We now extend this idea to multiply a monomial by a polynomial.
PRODUCTS OF POLYNOMIALSOBJECTIVESUpon completing this section you should be able to:
In the previous section you learned that the product A(2x + y) expands to A(2x) + A(y). Now consider the product (3x + z)(2x + y). Since (3x + z) is in parentheses, we can treat it as a single factor and expand (3x + z)(2x + y) in the same manner as A(2x + y). This gives us If we now expand each of these terms, we have Notice that in the final answer each term of one parentheses is multiplied by every term of the other parentheses.
Since  8x and 15x are similar terms, we may combine them to obtain 7x. In this example we were able to combine two of the terms to simplify the final answer. Here again we combined some terms to simplify the final answer. Note that the order of terms in the final answer does not affect the correctness of the solution.
POWERS OF POWERS AND SQUARE ROOTSOBJECTIVESUpon completing this section you should be able to:
We now wish to establish a second law of exponents. Note in the following examples how this law is derived by using the definition of an exponent and the first law of exponents. by the meaning of the exponent 3. Now by the first law of exponents we have In general, we note that This means that the answer will be
If we sum the term a b times, we have the product of a and b. Hence we see that Second Law of Exponents If a and b are positive integers and x is a real number, then In words, "to raise a power of the base x to a power, multiply the exponents." .
Note that when factors are grouped in parentheses, each factor is affected by the exponent. . Again, each factor must be raised to the third power. Using the definition of exponents, (5)^{2} = 25. We say that 25 is the square of 5. We now introduce a new term in our algebraic language. If 25 is the square of 5, then 5 is said to be a square root of 25. If x^{2} = y, then x is a square root of y.
. From the last two examples you will note that 49 has two square roots, 7 and  7. It is true, in fact, that every positive number has two square roots.
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The principal square root of a positive number is the positive square root. The symbol "" is called a radical sign and indicates the principal
Note the difference in these two problems. a. Find the square roots of 25.
For a. the answer is +5 and 5 since ( + 5)^{2} = 25 and (  5)^{2} = 25.
THE DIVISION LAW OF EXPONENTSOBJECTIVESUpon completing this section you should be able to correctly apply the third law of exponents. Before proceeding to establish the third law of exponents, we first will review some facts about the operation of division.
From (3) we see that an expression such as is not meaningful unless we know that y ≠ 0. In this and future sections whenever we write a fraction it will be assumed that the denominator is not equal to zero. Now, to establish the division law of exponents, we will use the definition of exponents.
In such an example we do not have to separate the quantities if we remember that a quantity divided by itself is equal to one. In the above example we could write
From the preceding examples we can generalize and arrive at the following law: Third Law of Exponents If a and b are positive integers and x is a nonzero real number, then
DIVIDING A MONOMIAL BY A MONOMIALOBJECTIVESUpon completing this section you should be able to simplify an expression by reducing a fraction involving coefficients as well as using the third law of exponents. We must remember that coefficients and exponents are controlled by different laws because they have different definitions. In division of monomials the coefficients are divided while the exponents are subtracted according to the division law of exponents. If no division is possible or if only reducing a fraction is possible with the coefficients, this does not affect the use of the law of exponents for division.
DIVIDING A POLYNOMIAL BY A MONOMIALOBJECTIVESUpon completing this section you should be able to divide a polynomial by a monomial. To divide a polynomial by a monomial involves one very important fact in addition to things we already have used. That fact is this: When there are several terms in the numerator of a fraction, then each term must be divided by the denominator.
DIVIDING A POLYNOMIAL BY A BINOMIALOBJECTIVESUpon completing this section you should be able to correctly apply the long division algorithm to divide a polynomial by a binomial. The process for dividing a polynomial by another polynomial will be a valuable tool in later topics. Here we will develop the technique and discuss the reasons why it works in the future. This technique is called the long division algorithm. An algorithm is simply a method that must be precisely followed. Therefore, we will present it in a stepbystep format and by example.
Solution Step 1: Arrange both the divisor and dividend in descending powers of the variable (this means highest exponent first, next highest second, and so on) and supply a zero coefficient for any missing terms. (In this example the arrangement need not be changed and there are no missing terms.) Then arrange the divisor and dividend in the following manner: Step 2: To obtain the first term of the quotient, divide the first term of the dividend by the first term of the divisor, in this case . We record this as follows: Step 3: Multiply the entire divisor by the term obtained in step 2. Subtract the result from the dividend as follows:
Step 4: Divide the first term of the remainder by the first term of the divisor to obtain the next term of the quotient. Then multiply the entire divisor by the resulting term and subtract again as follows:
This process is repeated until either the remainder is zero (as in this example) or the power of the first term of the remainder is less than the power of the first term of the divisor. As in arithmetic, division is checked by multiplication. We must remember that (quotient) X (divisor) + (remainder) = (dividend). To check this example we multiply (x + 7) and (x  2) to obtain x^{2} + 5x  14. Since this is the dividend, the answer is correct.
The answer is x  3. Checking, we find (x + 3)(x  3)
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