6/23/2015 A polynomial is an algebraic expression that includes addition, subtraction, multiplication, and whole number exponents, such as: 4x3 –3x2 + 7x + 5 A polynomial can not include • roots or fractional exponents • fractions with the variable on the bottom (or negative exponents) • variables in the exponent “Polynomial” means “many parts”. • The parts of a polynomial are called terms. • Each + or – starts a new term. • 4x3 –3x2 + 7x + 5 has four terms. 1 6/23/2015 How many terms are in each How many terms are in each of these polynomials? of these polynomials? • x2 – 2x + 3 • x2 –2x + 3 3 • 4n5 + 2n4 + 3n3 –2n2 • 4n5 + 2n4 + 3n3 –2n2 4 • 5x2y5 + 2x3y6 • 5x2y5 + 2x3y6 2 In the polynomial In the polynomial 4x3 + 2y2 – 5z + 2 4x3 + 2y2 – 5z + 2 • How many terms are there? • How many terms are there? 4 • What is the coefficient on the 1st • What is the coefficient on the 1st term? term? 4 • What is the coefficient on the 3rd • What is the coefficient on the 3rd term? term? -5 • What is the base on the 2nd term? • What is the base on the 2nd term? y • Which term is a constant? • Which term is a constant? 4th We often classify polynomials We often classify polynomials by the number of terms they by the number of terms they have. In particular … have. In particular … • Monomial • Monomial = 1 term • Binomial • Binomial = 2 terms • Trinomial • Trinomial = 3 terms 2 6/23/2015 Your book will sometimes ask you to write polynomials in You can have more than one standard form. This just variable in a polynomial. means re-arranging the terms so the exponents count down For instance 5x2y4 + 2xy3 is a from left to right. binomial where each term has 2 variables. The degree of a term tells how big that term is. What is the degree of each The degree is the sum of the term of this polynomial? exponents in a term. 5x2y2 –2x5 + 4x7y3 – 7x + 5 In 5x2y4 + 2xy3, the degree of the 1st term is 6 and the degree of the 2nd term is 4. The degree of a whole polynomial is the largest of the What is the degree of each degrees of its terms. term of this polynomial? So, the degree of 5x2y2 –2x5 + 4x7y3 – 7x + 5 5x2y4 + 2xy3 is 6. 4 5 10 1 0 The degree of 5x2y2 –2x5 + 4x7y3 – 7x + 5 is 10. 3 6/23/2015 What is the degree of What is the degree of 5n5p3 + 2n2p7 –4n4p3 + 5np6? 5n5p3 + 2n2p7 –4n4p3 + 5np6? 8 9 7 7 The overall degree is 9. It’s easy to add or subtract Special types of polynomials: polynomials. • Quadratic One variable • Just combine like terms. Degree = 2 • When you do that, you • Cubic One variable add the coefficients, and Degree = 3 leave the variables and coefficients alone. 4 6/23/2015 (5x2 + 3x – 2) + (3x2 –7x + 8) The variables and exponents =8x2 – 4x + 6 act like a label for the terms, which is why they don’t change when you add or subtract. Simplify: Simplify: (6x2y4 + 2x3y+ 3x2y2) (6x2y4 + 2x3y+ 3x2y2) + (5x3y – 2x2y4 + 2x2y2) + (5x3y – 2x2y4 + 2x2y2) = 4x2y4 + 7x3y + 5x2y2 Simplify: Simplify: (y3 –2y2 + 3y + 1) + (3y3 + 2y2 –5y + 2) (y3 –2y2 + 3y + 1) + (3y3 + 2y2 –5y + 2) 4y3 – 2y + 3 5 6/23/2015 Simplify: Simplify: (7n4 + 8n3 –5n2) – (2n4 + 3n3 + n2) (7n4 + 8n3 –5n2) – (2n4 + 3n3 + n2) 5n4 + 5n3 –6n2 Simplify: Simplify: (5x2 –3x + 5) –(2x2 + 4x – 1) (5x2 –3x + 5) –(2x2 + 4x – 1) 3x2 – 7x + 6 Simplify: Simplify: (7x2 + 2y2) – (4x2 + 2xy – 3y2) (7x2 + 2y2) – (4x2 + 2xy – 3y2) 3x2 – 2xy + 5y2 6.