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An expression such as 9p is a term; the number 9 is the coefficient, p is the variable, and 4 is the exponent. The expression p means p . p . p . p while p means p . p and so on. Terms having the same variable and the same exponent, such as 9x and -3x are like terms. Terms that do not have both the same variable and the same exponent, such as m and m are unlike terms. A polynomial is a term or a finite sum of terms in which all variables have whole number exponents, and no variables appear in denominators. Examples of polynomials include

5x + 2x + 6x, 8m + 9m n - 6mn + 3n , 10 p, and -9

The following properties of real numbers are useful for performing operations on polynomials.

PROPERTIES OF REAL NUMBERS

For all real numbers a, b, and c,

1. Commutative properties:

a + b = b + a

ab = ba

2. Associative properties

(a + b) + c = a + (b + c)

(ab)c = a(bc)

3. Distributive property

a(b + c) = ab + ac

EXAMPLE 1

Properties of Real Numbers

(a) 2 + x = x + 2 Commutative property of addition

(b) x.3 = 3x Commutative property of multiplication

(c) (7x)x = 7(x.x) = 7x Associative property of multiplication

(d) 3(x + 4) = 3x + 12 Distributive property

The distributive property is used to add or subtract polynomials. Only like terms may be added or subtracted. For example,

12y + 6y = (12 + 6)y = 18y and

-2m + 8m = (-2 + 8)m = 6m but the polynomial 8y + 2y cannot be further simplified. To subtract polynomials, use the facts that -(a+b)=-a-b and -(a-b)=-a+b In the next example, we show how to add and subtract polynomials.