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Exploring Monomials and Polynomials in Algebra

Learn about monomials, polynomials, and properties of powers in algebra. Explore examples and practice problems for a better understanding of these fundamental concepts in mathematics.

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Exploring Monomials and Polynomials in Algebra

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  1. Chapter 5 Exploring Polynomials & Radical Expressions By Wendi Kelson

  2. 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

  3. 5-1 Monomials (cont.) • 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 y8 y*y*y*y*y*y*y*y y3 y*y*y = = y8-3 = y5

  4. 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. 5-1 Practice • What is the degree of 7x2y11? 2. Simplify x2 * x5. 3. Simplify y9/y5. Answers: 1) 13 2) x7 3) y4

  6. 5-1 Practice (cont.) • Simplify (z4)3. • Simplify (2x)5. • Simplify (2/5)-3. Answers: 1) z12 2) 32x5 3) 125/8

  7. 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

  8. 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.

  9. 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.

  10. 5-9 Complex Numbers The Complex Number System

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