7.1 An Introduction to Polynomials Objectives: Identify, evaluate, add, and subtract polynomials. Classify polynomials, and describe the shapes of their graphs. subtract polynomials. Classify polynomials, and describe the shapes of their graphs. Standard: S. Analyze properties and relationships of polynomials.

A monomial is a numeral, a variable, or the product of a numeral and one or more variables. A monomial with no variables, such as –1 or, is called a constant. A coefficient is the numerical factor in a monomial. The degree of a monomial is the sum of the exponents of its variables. A polynomial is a monomial or a sum of terms that are monomials. A polynomial with two terms is a binomial. A polynomial with three terms is a trinomial. The degree of a polynomial is the same as that of its term with the greatest degree.

Ex 1. Classify each polynomial by degree and by number of terms. First, combine Like terms with the same letter and exponent power. Second, what is the highest exponent and name it by the degree. Third, count how many terms there are and name it. a.2x 3 - 3x + 4x 3 b. –2x 3 + 3x 4 + 2x c. x x - 2x 3 d. 3x – x 3 – 6x 5 e. 5x + 2x 3 + 4x 2 f. x 5 – 4x 3 - x 5 + 3x 2 + 4x 3 6x 3 – 3xCubic Binomial3x 4 + 5Quartic Binomial Cubic Polynomial 2x 3 – 6x 5 + 2Quintic Trinomial Cubic Trinomial3x 2 Quadratic Monomial

Adding and Subtracting Polynomials To add and subtract polynomials, combine like terms. To add and subtract polynomials, combine like terms. The standard form of a polynomial expression is written with the exponents in descending order of degree. The standard form of a polynomial expression is written with the exponents in descending order of degree.

Ex 2. Find the sum and write it from highest to lowest power. a. (-2x 2 – 3x 3 + 5x + 4) + (-2x 3 + 7x – 6) a. (-2x 2 – 3x 3 + 5x + 4) + (-2x 3 + 7x – 6) -5x 3 – 2x x – 2 -5x 3 – 2x x – 2 b. (2x 4 + 4x 3 + 5x - 2) + (-2x 4 – 7x 2 + 8x – 10) b. (2x 4 + 4x 3 + 5x - 2) + (-2x 4 – 7x 2 + 8x – 10) 4x 3 – 7x x – 12 4x 3 – 7x x – 12 c. (6x 3 + 3x 2 – 4) + (2x 3 – 5x 2 – 3x – 10) c. (6x 3 + 3x 2 – 4) + (2x 3 – 5x 2 – 3x – 10) 8x 3 – 2x 2 – 3x – 14 8x 3 – 2x 2 – 3x – 14

Ex 3. Find the difference and write it from highest to lowest power. a. (-6x 3 – 6x 2 + 7x – 1) – ( 3x 3 – 5x 2 – 2x + 8) a. (-6x 3 – 6x 2 + 7x – 1) – ( 3x 3 – 5x 2 – 2x + 8) -9x 3 – x 2 + 9x – 9 -9x 3 – x 2 + 9x – 9 b. ( 3x 3 – 12x 2 – 5x + 1) – (-x 2 + 5x + 8) b. ( 3x 3 – 12x 2 – 5x + 1) – (-x 2 + 5x + 8) 3x 3 – 11x 2 – 10x – 7 3x 3 – 11x 2 – 10x – 7 c. ( 5x 2 – 6x – 11) – (-8x 3 + x 2 + 2) c. ( 5x 2 – 6x – 11) – (-8x 3 + x 2 + 2) 8x 3 + 4x 2 – 6x – 13 8x 3 + 4x 2 – 6x – 13

Graphing Polynomial Functions A polynomial function is a function that is defined by a polynomial expression. Ex 4. Graph each function. Describe its general shape. Ex 4. Graph each function. Describe its general shape. a. P(x) = 3x 3 - 5x 2 - 2x + 1 Cubic – S shape Cubic – S shape b. Q(x) = x 4 - 8x 2 Quartic – W Shape Quartic – W Shape c. P(x) = -3x 3 - 2x 2 + 2x - 1 Cubic – S Shape Cubic – S Shape

Examine the shapes of the linear, quadratic, cubic, and quartic functions shown below. Linear Quadratic Cubic Quartic Line U S W Line U S W

Writing Activities 1). Describe 2 different ways to classify polynomials. Include examples. 2). What are the degree and the leading coefficient of Explain. 3). Which of the following are like terms? Explain. Explain.

Review Of Introduction To Polynomials