Chapter 5

Exploring Polynomials & Radical Expressions

By Wendi Kelson

5-1 Monomials

• A monomial is a number, variable, or a number and 1 or more variables multiplied together.– Examples: 10, x, 13y, ¼x3y4

• A constant is a monomial that has just numbers & no variable.

• The coefficient is the number in front of the variable.– Example: 5x → coefficient is 5

• The degree of a monomial is the sum of the exponents.– Example: 18x6y3 → degree is 6+3 = 9

• A power is an expression in the form xn.

• Multiplying Powers → am * an = am+n

– Example: 52 * 53 = 52+3 = 55

• Dividing Powers → am/an = am-n

– Example:

• When c is a nonzero number, c0 = 1

5-1 Monomials (cont.)

y8 y*y*y*y*y*y*y*y

y3 y*y*y= = y8-3 = y5

5-1 Monomials (cont.)Properties of Powers

• Power of a Power → (am)n = am*n

– Example: (52)3 = 52*3 = 56

• Power of a Product → (ab)m = am * bm

– Example: (5*x)3 = 53 * x3 = 125x3

• Power of a Quotient → (a/b)n = an/bn and (a/b)-n = (b/a)n

– Examples: (5/3)2 = 52/32 = 25/9

(5/3)-2 = (3/5)2 = 32/52 = 9/25

5-1 Practice

1. What is the degree of 7x2y11?

2. Simplify x2 * x5.

3. Simplify y9/y5.

Answers: 1) 13 2) x7 3) y4

5-1 Practice (cont.)

1. Simplify (z4)3.

2. Simplify (2x)5.

3. Simplify (2/5)-3.

Answers: 1) z12 2) 32x5 3) 125/8

5-2 Polynomials

• A polynomial is a monomial or the sum of 2 or more monomials.– Example: 2x2 + 3x + 1

• The terms of a polynomial are the monomials that make it up.

• Like terms in a polynomial are two terms that are the same except for their coefficients.– Example: In the equation 2x2 + 2x + x + 1, 2x and x

are like terms and can be combined to 3x. → 2x2 + 3x +1

5-2 Polynomials

• A polynomial with 2 unlike terms is called a binomial, and a polynomial with 3 unlike terms is called a trinomial.

• The degree of a polynomial is the degree of the monomial with the biggest degree.– Example: The degree of 3x2 – 10x – 8 is 2.

5-9 Complex Numbers

• Imaginary unit: i = √-1• A pure imaginary number is of the form bi, while

the form a ± bi is a nonpure imaginary number.• A complex number is a number in the form of

a ± bi, where a can be either zero or a real number.

5-9 Complex Numbers

(a + bi)ComplexNumbers

(b = 0)Real

Numbers

RationalNumbers

IrrationalNumbers

(b ≠ 0)ImaginaryNumbers

(a = 0) Pure

ImaginaryNumbers

(a ≠ 0)Nonpure

ImaginaryNumbers

The Complex Number System