5 5 Polynomials Goals 1 To identify terms

5. 5 Polynomials • Goals: 1. To identify terms, coefficients, and factors of a term. 2. To simplify a poly by collecting like terms 3. To identify the degree of the poly

Monomial Polynomials A Polynomial is a monomial, or a sum of monomials 3 x 2 + 2 x - 5 Trinomial Binomial 5 x + 3

To determine if an expression is a polynomial: First check that every term is a monomial. If every term is a monomial then the expression is a polynomial

Is the term a polynomial? If so, say whether it is a monomial, binomial, or trinomial. y 2 + 2 y Binomial, since it has two TERMS “terms” are added, “factors” are multiplied

Is the term a polynomial? If so, say whether it is a monomial, binomial, or trinomial. No “terms” are added, “factors” are multiplied

Is the term a polynomial? If so, say whether it is a monomial, binomial, or trinomial. YES, trinomial since it has 3 terms Numeric factor of a term is called the coefficient.

To simplify by collecting “like terms” have the same exact variables to the same exact power. Only the coefficients may be different. 7 m 2 – 2 m 2 = (7 – 2)m 2 = 5 m 2

To simplify by collecting “like terms” have the same exact variables to the same exact power. Only the coefficients may be different. 3 x 5 y 4 – 6 y 4 – x 5 y 4 + 2 y 4 4 5 4 +(– 6 + 2) y = (3 – 1)x y = 2 x 5 y 4 - 4 y 4

Simplify by collecting like terms: 2 m 4 – 4 m 3 + 3 m 3 – 1 2 m 4 – m 3 – 1

Simplify by collecting like terms: 3 m 5 – 3 m 2 + m – 1 (no like terms)

Simplify by collecting like terms: 3 x 5 y 4 – 6 y 4 + m 3 – x 5 y 4 2 x 5 y 4 – 6 y 4 + m 3 9 th deg 4 th deg 3 rd deg The Degree of a term is the sum of the exponents of the variables.

Simplify by collecting like terms: 3 x 5 y 4 – 6 y 4 + m 3 – x 5 y 4 2 x 5 y 4 – 6 y 4 + m 3 This is a 9 th degree polynomial. The Degree of a term is the sum of the exponents of the variables. The Degree of a polynomial is the highest degree of any term.

Identify the degree of each term and the degree of the polynomial. - 63 x 2 y 3 – 16 xy 5 + y 4 5 th deg 6 th deg 4 th deg This is a 6 th degree polynomial. The Degree of a term is the sum of the exponents of the variables. The Degree of a polynomial is the highest degree of any term.

Identify the degree of each term and the degree of the polynomial. 9 x 6 y 5 – 7 x 4 y 3 + 3 xy 4 11 th deg 7 th deg 5 th deg This is a 11 th degree polynomial. The Degree of a term is the sum of the exponents of the variables. The Degree of a polynomial is the highest degree of any term.

9 x 6 y 5 – 7 x 4 y 3 + 3 xy 4 • Leading term is the term that has the highest degree in the poly • Ex: 9 x 6 y 5 • Leading coefficient is the coefficient of the leading term. • Ex: 9

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