Polynomials What is a polynomial A polynomial in

What is a polynomial? A polynomial in one variable is any expression that can

What does that mean? • • Exponents of variables can’t be negative (4 x-2)

Let’s practice: with your seat-partner, determine which are polynomials and label them 1 2

How do we talk about polynomials? • One thing we talk about with polynomials

Let’s practice: with your seat-partner, write down how many terms each polynomial has, and

What degree is it? • Another way we classify polynomials is by the degree

Let’s practice: with your seat-partner, write down the degree of each polynomial 1 2

Standard form? • Polynomials are said to be in “standard form” if they are

Let’s practice: with your seat-partner, write down which polynomials are in standard form, and

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Polynomials!

What is a polynomial? A polynomial in one variable is any expression that can be written in the form anxn + an-1 xn-1 + ··· + a 1 x 1 + a 0 Where x is a variable, the exponents are nonnegative integers, the coefficients are real numbers, and an ≠ 0. Each monomial being added to make the polynomial is a term.

What does that mean? • • Exponents of variables can’t be negative (4 x-2) Exponents of variables can’t be fractions (4 x 1/2) No taking the root of variables (4√x) Coefficients have to be real numbers No dividing by variables (4/x) The pieces between plus signs are called terms In standard form, terms are written in order from biggest degree to smallest degree

Let’s practice: with your seat-partner, determine which are polynomials and label them 1 2 3 5 x 2 + 4 4 x - √ 9 Polynomial, two terms, binomial, 2 nd degree binomial, 1 st degree 4 - √x Not a polynomial 4 5 6 4 x 6 + 12 x 4 - 3 Polynomial, three terms, trinomial, 6 th degree 6 x 9 + (4/x) – 10 Not a polynomial 5 x 4 + 12 x 9 + 10 x 5 – x Polynomial, four terms, 9 th degree 7 8 9 16 x-4 + x – 8 Not a polynomial (6 x + 3)(4 x – 12) 24 x 2 + 84 x - 36 Polynomial, have to multiply it out to see 3 terms, trinomial, 2 nd degree 168 Polynomial, one term, monomial, 0 th degree

How do we talk about polynomials? • One thing we talk about with polynomials is how many terms they have • Monomial means 1 term: 4 x 2 • Binomial means 2 terms: 4 x 2 – 4 • Trinomial means 3 terms: 4 x 2 + 4 x – 4 • Polynomial means many terms: 4 x 5 + 4 x 4 + 4 x 3 + 4 x 2 + 4 x + 4

Let’s practice: with your seat-partner, write down how many terms each polynomial has, and label monomials, binomials, and trinomials 1 2 3 5 x 2 + 4 4 x - √ 9 Polynomial, two terms, binomial, 2 nd degree binomial, 1 st degree 4 - √x Not a polynomial 4 5 6 4 x 6 + 12 x 4 - 3 Polynomial, three terms, trinomial, 6 th degree 6 x 9 + (4/x) – 10 Not a polynomial 5 x 4 + 12 x 9 + 10 x 5 – x Polynomial, four terms, 9 th degree 7 8 9 16 x-4 + x – 8 Not a polynomial (6 x + 3)(4 x – 12) 24 x 2 + 84 x - 36 Polynomial, have to multiply it out to see 3 terms, trinomial, 2 nd degree 168 Polynomial, one term, monomial, 0 th degree

What degree is it? • Another way we classify polynomials is by the degree of the polynomial • This is determined by the highest power an exponent has • 0 th degree: 658 x 0 Normally written: 658 • 1 st degree: 4 x 1 + 1 Normally written: 4 x + 1 • 2 nd degree: 4 x 2 + 1 • 3 rd degree: 4 x 3 + 5 x 2 + 1 • And so on

Let’s practice: with your seat-partner, write down the degree of each polynomial 1 2 3 5 x 2 + 4 4 x - √ 9 Polynomial, two terms, binomial, 2 nd degree binomial, 1 st degree 4 - √x Not a polynomial 4 5 6 4 x 6 + 12 x 4 - 3 Polynomial, three terms, trinomial, 6 th degree 6 x 9 + (4/x) – 10 Not a polynomial 5 x 4 + 12 x 9 + 10 x 5 – x Polynomial, four terms, 9 th degree 7 8 9 16 x-4 + x – 8 Not a polynomial (6 x + 3)(4 x – 12) 24 x 2 + 84 x - 36 Polynomial, have to multiply it out to see 3 terms, trinomial, 2 nd degree 168 Polynomial, one term, monomial, 0 th degree

Standard form? • Polynomials are said to be in “standard form” if they are in order from the highest degree term to the lowest. • In standard form • x 4 + x 3 + x 2 + x 1 + x + n • Not in standard form • x 3 + x 4 + x 1 + x 2 + n + x

Let’s practice: with your seat-partner, write down which polynomials are in standard form, and which are not 1 2 3 5 x 2 + 4 4 x - √ 9 Polynomial, two terms, binomial, 2 nd degree binomial, 1 st degree 4 - √x Not a polynomial 4 5 6 4 x 6 + 12 x 4 - 3 Polynomial, three terms, trinomial, 6 th degree 6 x 9 + (4/x) – 10 Not a polynomial 5 x 4 + 12 x 9 + 10 x 5 – x Polynomial, four terms, 9 th degree 7 8 9 16 x-4 + x – 8 Not a polynomial (6 x + 3)(4 x – 12) 24 x 2 + 84 x - 36 Polynomial, have to multiply it out to see 3 terms, trinomial, 2 nd degree 168 Polynomial, one term, monomial, 0 th degree

Let’s practice: with your seat-partner, write down which polynomials are in standard form, and which are not 1 standard 2 standard 5 x 2 + 4 4 x - √ 9 Polynomial, two terms, binomial, 2 nd degree binomial, 1 st degree 4 standard 4 x 6 + 12 x 4 - 3 Polynomial, three terms, trinomial, 6 th degree 7 16 x-4 + x – 8 Not a polynomial 5 6 x 9 + (4/x) – 10 Not a polynomial 8 not standard (6 x + 3)(4 x – 12) 24 x 2 + 84 x - 36 Polynomial, have to multiply it out to see 3 terms, trinomial, 2 nd degree 3 4 - √x Not a polynomial 6 not standard 5 x 4 + 12 x 9 + 10 x 5 – x Polynomial, four terms, 9 th degree 9 standard 168 Polynomial, one term, monomial, 0 th degree