Introduction to Polynomials

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7.1 An Introduction to Polynomials

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Presentation transcript:

Introduction to Polynomials 24 February 2011

What is a polynomial? polynomial

Definition of Polynomial A polynomial is algebraic expression that can be written in the form anxn + an-1xn-1 + … + a2x2 + a1x1 + a0 An equation or an expression with a single variable raised to (usually many) powers All exponents are whole numbers an ≠ 0

Key Features of Polynomials Degree (n, greatest exponent in polynomial) anxn + an-1xn-1 + … + a2x2 + a1x1 + a0 Leading Coefficient (an) Constant (the term without a variable)

Notes About Polynomials Exponents can be skipped! Ex. 8x8 – x4 + 2x – 1 Written in numerical order from greatest exponent to least exponent Ex. 2x3 + 7x8 – 14x2 – 12 + 5x5 Rewrite: 7x8 + 5x5 + 2x3 – 14x2 – 12

Classifying Polynomials Classified according to key features Degree Leading Coefficient Constant Degree especially important!

Naming Polynomials Named according to their degree Ex. 3x11 + 7x5 – 11x3 + 2x2 + 14x – 63 11th degree polynomial Some polynomials have special names: 0 degree (just a constant term): Constant 1st degree: Linear 2nd degree: Quadratic 3rd degree: Cubic 4th degree: Quartic 5th degree: Quintic

Examples: x3 Degree: Name: 4x3 + 10 + 6x7 + 5x2 – 7x Rewrite:

Examples: 0x9 + 2x6 + 3x7 + x8 – 2x – 4 Rewrite: x8 + 3x7 + 2x6 – 2x – 4 Degree: 8 Name: 12 Degree:

Your Turn: On the “Introduction to Polynomials” Handout, find the table labeled “Classifying Polynomials”. Identify the degree, leading coefficient, and constant of each of the polynomials in the table.

Your Turn: Go back to the “Classifying Polynomials” table. Name each polynomial.

Multiplication of Polynomials Monomial by anything = Distribute Binomial by binomial = FOIL Multiplying with trinomials (or larger) = Variation of FOIL Make sure you distribute each term in the 1st set of parentheses to every term in the 2nd set of parentheses

Example #1 (– 4x2 – 5x – 1)(4x2 – 6x – 2)

Example #2 (x – 2)3

Your Turn: On the “Introduction to Polynomials” handout, complete problems 1 – 14. Remember to simplify and to rewrite the polynomials from greatest exponent to least exponent!