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What's the Deal? <ul><li>Be able to determine the degree of a polynomial. </li></ul><ul><li>Be able to classify a polynomial. </li></ul><ul><li>Be able to write a polynomial in standard form. </li></ul><ul><li>Be able to add and subtract polynomials </li></ul>
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Vocabulary Monomial: A number, a variable or the product of a number and one or more variables. Polynomial: A monomial or a sum of monomials. Binomial: A polynomial with exactly two terms. Trinomial: A polynomial with exactly three terms. Coefficient: A numerical factor in a term of an algebraic expression.
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Vocabulary Degree of a monomial: The sum of the exponents of all of the variables in the monomial. Degree of a polynomial in one variable: The largest exponent of that variable. Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order.
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Polynomials in One Variable <ul><li>A polynomial is a monomial or the sum of monomials </li></ul><ul><li>Each monomial in a polynomial is a term of the polynomial. </li></ul><ul><ul><li>The number factor of a term is called the coefficient. </li></ul></ul><ul><ul><li>The coefficient of the first term in a polynomial is the lead coefficient . </li></ul></ul><ul><li>A polynomial with two terms is called a binomial. </li></ul><ul><li>A polynomial with three terms is called a trinomial. </li></ul>
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Polynomials in One Variable <ul><li>The degree of a polynomial in one variable is the largest exponent of that variable. </li></ul>A constant has no variable. It is a 0 degree polynomial. This is a 1 st degree polynomial. 1 st degree polynomials are linear . This is a 2 nd degree polynomial. 2 nd degree polynomials are quadratic . This is a 3 rd degree polynomial. 3 rd degree polynomials are cubic.
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Classify the polynomials by degree and number of terms. Examples Polynomial a. b. c. d. Degree Number of Terms Classify by number of terms Zero 1 Monomial First 2 Binomial Second 2 Binomial Third 3 Trinomial
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Standard Form <ul><li>To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. </li></ul><ul><li>The leading coefficient , the coefficient of the first term in a polynomial written in standard form, should be positive. </li></ul>
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Examples Write the polynomials in standard form. Remember: The lead coefficient should be positive in standard form. To do this, multiply the polynomial by –1 using the distributive property.
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Practice Write the polynomials in standard form and identify the polynomial by degree and number of terms. 1. 2.
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Problem 1 This is a trinomial. The trinomial’s degree is 3.
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Problem 2 This is a 2 nd degree, or quadratic, trinomial.
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Find the Sum <ul><li>Add (x 2 + x + 1) to (2x 2 + 3x + 2) </li></ul><ul><li>You might decide to add like terms as the next slide demonstrates. </li></ul>
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Add Like Terms + 2x 2 + 3x + 2 x 2 + x + 1 = 3x 2 + 4x +3 Or you could add the trinomials in a column
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Just like adding like-terms + 2x 2 + 3x + 2 x 2 + x + 1 3x 2 + 4x +3 Start with the trinomials in a column + 2x 2 + 3x + 2 Combine the trinomials going down
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Problem #2 <ul><li>Try one. </li></ul><ul><li>(3x 2 +5x) + (4 -6x -2x 2 ) </li></ul><ul><li>Make sure you put the polynomials in standard form and line them up by degree. </li></ul>
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(3x 2 +5x) + (4 -6x -2x 2 ) 3x 2 +5x -2x 2 -6x + 4 + 0 x 2 -x + 4 It might be helpful to use a zero as a placeholder.
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Find the difference - (3x 2 - 2x + 3) x 2 + 2x - 4 -2x 2 + 4x - 7 Start with the trinomials in a column - (3x 2 - 2x + 3) The negative sign outside of the parentheses is really a negative 1 that is multiplied by all the terms inside. - 3x 2 + 2x - 3)
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Try One. - (10x 2 + 3x + 2) 12x 2 +5x + 11 2x 2 + 2x + 9 Reminder: Start with the trinomials in a column - (10x 2 + 3x + 2) The negative sign outside of the parentheses is really a negative 1 that is multiplied by all the terms inside. - 10x 2 - 3x - 2
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Special Thanks to Public Television Station KLVX for the basic outline of the first 12 slides of this presentation
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