Polynomials

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 pmeans 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 + 9mn - 6mn + 3n, 10 p, and -9

Adding and Subtracting Polynomials

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.