Notes 7-3: Polynomials Remember from 7-1: A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. A monomial may be a constant or a single variable. I. Identifying Polynomials

A polynomial is a monomial or a sum or difference of monomials.

Some polynomials have special names. A binomial is the sum of two monomials. A trinomial is the sum of three monomials. • Example: State whether the expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. Expression Polynomial? Monomial, Binomial, or Trinomial? 2x - 3yz Yes, 2x - 3yz = 2x + (-3yz), the binomial sum of two monomials 8n3+5n-2 No, 5n-2 has a negative None of these exponent, so it is not a monomial -8 Yes, -8 is a real number Monomial

4a2 + 5a + a + 9 Yes, the expression simplifies Monomial to 4a2 + 6a + 9, so it is the sum of three monomials II. Degrees and Leading Coefficients The terms of a polynomial are the monomials that are being added or subtracted.

The degree of a polynomial is the degree of the term with the greatest degree.

The leading coefficient is the coefficient of the variable with the highest degree. Find the degree and leading coefficient of each polynomial Polynomial Terms Degree Leading Coefficient 5n2 5n 2 2 5

-4x3 + 3x2 + 5 -4x2, 3x2, 3 -4 5 -a4-1 -a4, -1 4 -1 III. Ordering the terms of a polynomial

The terms of a polynomial may be written in any order. However, the terms of a polynomial are usually arranged so that the powers of one variable are in descending (decreasing, large to small) order. Examples: Arrange the terms of each polynomial so that the powers of x are in descending order.

A. 7x2 + 2x4 - 11 B. 2xy3 + y2 + 5x3 – 3x2y

2x4 + 7x2 – 11x 2x1y3 + y2 + 5x3 – 3x2y1

5x3 -3x2y + 2xy3 +y2 Arrange the terms of each polynomial so that the powers of x are in descending order.

C. 6x2 + 5 – 8x -2x3

-2x3 + 6x2– 8x + 5

D. 3a3x2 – a4 + 4ax5 + 9a2x

4ax5 + 3a3x2 + 9a2x – a4 You Try! Arrange the terms of each polynomial so that the powers of x are in descending order.

E. 3x2y4 + 2x4y2 – 4x3y + x5 – y2

x5+ 2x4y2– 4x3y + 3x2y4 -y2