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Introduction to Polynomials

Learn about polynomials and their classifications based on number of terms and degree. Understand how to identify the degree and classify polynomials by name. Explore standard form and how to classify polynomials based on their characteristics.

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Introduction to Polynomials

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  1. BELL-WORK

  2. BELL-WORK

  3. Polynomials The expression polynomial is the general name given to mathematical terms. Ex. 1, x + 8, 3x2 – 2x + 6 Polynomials are classified by number of terms and by degree. Definitions by number of terms Monomial: an expression that is a number, a variable, or a product of a number and one or more variables, Ex.:1, 2, 3, x, y2, z12, 2x3, 5y, 12x2y5

  4. Polynomials Definitions by number of terms Binomial: an expression that is the sum or difference of two monomials. Ex.: x + 1, 3x2 – 5x4 Trinomial: an expression that is the sum or difference of three monomials. Ex.: 3x2 + 2x – 1

  5. Polynomials Definitions by number of terms Polynomial: is a monomial or the sum or difference of two or more monomials. The term is usually used to refer to the sum or difference of more than three monomials. Ex.: 6x5 – 5x4 + 3x3 – 2x2 – x + 1

  6. Polynomials Name each polynomial based on the number of terms: (a) 5x2 + 2x + 1: Trinomial (b) 3z – 2: Binomial

  7. The Degree of a Monomial The degree of a monomial is the sum of the exponents of its variables. Ex. x2:2nd degree 5x: 1st degree 6: 0 degree 7x3: 3rd degree 9y4: 4th degree 12x5y3:8th degree What is the degree of 8xy: 2nd degree -7y4z: 5th degree 11: 0 degree Note: Zero has no degree.

  8. The Degree of a Monomial A term must have a degree that is a positive number. A term with a degree that is not positive is not considered a polynomial. Ex: 6x2y-7 Not a polynomial 6x7y-2 Not a polynomial

  9. The Degree of a Monomial Names by degree: Constant: the name given to a monomial with degree zero. Ex.: 2 Linear: the name given to a monomial with degree one. Ex. 2x Quadratic: the name given to a monomial with degree two. Ex.: 2x2 Cubic: the name given to a monomial with degree three. Ex. 2x3

  10. The Degree of a Monomial Names by degree: Quartic: the name given to a monomial with degree four. Ex. 2yx3 Quintic: the name given to a monomial with degree five. Ex. 2x2y2z After the 5th degree use nth degree. Ex. 2x6 6th degree Commit these terms to memory!

  11. The Degree of a Polynomial The degree of a polynomialis the highest degree of any of its monomials. What is the degree of the following polynomials: 3x4 + 5x3 – 12x2 – x + 1 4th degree = quartic 2x – x8 – 3x3 + x2 8th degree 2 + y – 7y3 + 15y5 5th degree = quartic On your HW, be sure to give the name by number and word!

  12. Standard Form of a Polynomial Before a polynomial can be classified it must first be written in standard form. A polynomial is in standard form when each monomial is written in order with decreasing degrees. Ex.: 3x4 + 5x3 – 12x2 – x + 1 Are the following polynomials in standard form? If not write them in standard form. x8 – 3x3 + x2 + 2x Yes! 2 + y – 4y3 + 15y5 –3y3 No! 15y5 – 7y3 + y + 2 Note: Like-terms must be combined to be considered standard form.

  13. Classifying Polynomials To classify a polynomial, first give the name according to the degree, and then give the name according to the number of terms. Note: Be sure to have the polynomial written in standard form. Ex.: 7x + 4 Standard form? Yes By degree it is linear By number of terms it is binomial Classification: Linear binomial

  14. Classifying Polynomials 2p + 3p2 – 1 Standard form? No 3p2 + 2p – 1 By degree it is quadratic By number of terms it is trinomial Classification: Quadratic trinomial

  15. Classifying Polynomials 4z3 Standard form? Yes By degree it is cubic By number of terms it is monomial Classification: Cubic monomial 9t4 + 11t Standard form? Yes By degree it is quartic By number of terms it is binomial Classification: Quartic binomial

  16. Classifying Polynomials 5 Standard form? Yes By degree it is constant By number of terms it is monomial Classification: Constant monomial Classify 2x – 3 + 8x2 Quadratic trinomial

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