Algebra

C

Special Definitions

Any statement that contains the equality relation (=), such as 3x = 9, is called an equation. An equation is called an identity if the equality is true for all values of its variables; if the equation is true for some values of its variables and false for others, the equation is conditional. The equation x + 0 = x, for example, is an identity while 3x = 9 is conditional because it is only true when x = 3. A term is any algebraic expression consisting only of products of constants and variables; 2x, -a, and ‚s⁴x are all examples of terms. The numerical part of a term is called its coefficient. The coefficients of each term above are, respectively, 2, -1, and ‚.

An expression containing one term, such as 2x³, is called a monomial. An expression involving the addition or subtraction of two terms, as in 2x² + 3x, is called a binomial, while an expression with three terms, such as 4x⁵ – x⁴ + 7x, is known as a trinomial. Polynomial is the general name for expressions in which any number of terms are added or subtracted. The degree of a polynomial refers to the largest exponent of the variables in the polynomial. For example, if the largest exponent of a variable is 3, as in ax³ + bx², the polynomial is said to be of degree 3. Similarly, the expression xⁿ + x^n-1 + x^n-2 is of degree n.

A linear equation with one variable is a polynomial equation of degree one—that is, of the form ax + b = 0. These are called linear equations because graphing these equations results in straight lines. A quadratic equation in one variable is a polynomial equation of degree two—that is, of the form ax² + bx + c = 0.

An indeterminate equation, such as x² + y² = z², involves multiple unknowns.

---

---

A prime number is any integer (the counting numbers: 1, 2, 3, …; their negatives; and zero) that can be evenly divided only by itself and by the number 1 or the number -1. Thus, 2, 3, 5, 7, 11, 13, 17, and 19 are all prime numbers.

A factor of a number is any integer by which the number can be divided evenly, with no remainder. The factors of 6, for example, are 1, 2, 3, and 6, because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1. The prime factors of any number are those factors to which it can be reduced such that the number is expressed only as the product of primes and their powers. For example, the prime factors of 6 are 2 and 3. Similarly, because 60 = 2² × 3 × 5, the prime factors of 60 are 2, 3, and 5.

III

Putting Polynomial Equations Into Solvable Forms

Solving a polynomial equation usually requires altering its form. The most common tools used to manipulate equations into solvable form are the laws of arithmetic, factoring, and the finding of least common multiples.

A

Laws of Polynomial Arithmetic

In manipulating polynomials, the usual laws of the arithmetic of numbers hold. This section lays out those laws. Normal arithmetic is limited to the set of rational numbers (positive and negative whole numbers and infinitely repeating decimals). Algebra and geometry can include irrational numbers (infinite decimals that do not repeat), such as pi (symbol p) and the square root of 2 (abbreviated √2). The set of all rational and irrational numbers taken together constitutes the set of real numbers. See also Number (mathematics).

A

1

Laws of Addition

1. The sum of any two real numbers a and b is also a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number.

2. No matter how terms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.

3. Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.

4. Given any real number a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.

5. No matter in what order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition.

Any set of numbers obeying laws 1 through 4 is said to form a group. If the set also obeys law 5, it is said to be an Abelian, or commutative, group. Integers and real numbers are both Abelian groups. Since subtraction can be treated as the addition of negative numbers (3 – 4 is the same as 3 + -4), these laws also apply to subtraction.

A

2

Laws of Multiplication

Laws similar to those for addition also apply to multiplication. Since powers are a special case of multiplication, these laws cover powers as well.

1. The product of any two real numbers a and b is also a real number, denoted a·b or ab.

2. No matter how terms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.

3. Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.

4. Given any nonzero real number a, there is a number (a^-1), or (1/a), called the multiplicative inverse, such that a(a^-1) = (a^-1)a = 1.

5. No matter in what order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero (because division by zero is impossible), forms such a commutative group under multiplication. Law number 4 allows the laws of multiplication to be extended to division, since dividing by a number is the same as multiplying by its inverse.

	More from Encarta
	•  Presidential Myths Quiz •  Coffee break: Recharge your brain