6/24/2016 4:17 AM6.1 - Polynomials1 Section 6.1 P O L Y N O M I A L S

6/24/2016 4:17 AM6.1 - Polynomials2 R EVIEW Coefficient: The number in front of the variable. Degree: The sum of the exponents of the variables. ADD THE TOTAL EXPONENTS Term: Parts of an expression that is added or subtracted Constant Term: The number at the end of the equation that does not have the variable. Standard Form: Descending degree form in equation.

6/24/2016 4:17 AM6.1 - Polynomials3 D EGREE E XAMPLES Identify the total degrees of each monomial. A. z 6 Identify the exponent. B. 5.6 The degree is 6. z6z6 5.6 = 5.6x 0 Identify the exponent. The degree is 0. C. 8xy 3 Add the exponents. The degree is 4. 8x1y38x1y3 D. a 2 bc 3 a2b1c3a2b1c3 Add the exponents. The degree is 6.

6/24/2016 4:17 AM6.1 - Polynomials4 D EFINITIONS Monomial: A number or a product of numbers and variables with whole number exponents. Binomial: A polynomial with two terms Trinomial: Polynomial with three terms Polynomial: A monomial or the sum or difference of monomials. WHOLE NUMBER DEGREES

6/24/2016 4:17 AM6.1 - Polynomials5 I S IT A P OLYNOMIAL ? EExamples of polynomials: ––3x 4 00.15x 101 . EExamples of non-polynomials: .. 1 2 a7a7 |2b 3 – 6b| m 0.75 – m 1 2

6/24/2016 4:17 AM6.1 - Polynomials6 R ANKS OF P OLYNOMIALS Classifying by degree:

6/24/2016 4:17 AM6.1 - Polynomials7 R ANKS OF P OLYNOMIALS Degree Terms 1 st Linear (x) 2 nd Quadratic (x 2 ) 3 rd Cubic (x 3 ) 4 th Quartic (x 4 ) 5 th & beyond 5 th degree, 6 th degree, etc… 1Monomial 2Binomial 3Trinomial 4 and beyond 4 th term, 5 th term, etc…

6/24/2016 4:17 AM6.1 - Polynomials8 C LASSIFYING P OLYNOMIALS Refresh: Standard Form: Descending degree form in equation. Rewrite each polynomial in standard form. Then identify the leading coefficient, HIGHEST degree, and number of terms. Label the polynomial – 5x 2 + 4x –5x 2 + 4x + 3 Write terms in descending order by degree. Leading coefficient: –5 Terms: 3 Highest Degree: 2 Label: Quadratic Trinomial

6/24/2016 4:17 AM6.1 - Polynomials9 C LASSIFYING P OLYNOMIALS Refresh: Standard Form: Descending degree form in equation. Rewrite each polynomial in standard form. Then identify the leading coefficient, HIGHEST degree, and number of terms. Label the polynomial. 2. 5x 2 – 4 + 8x 4 + x 8x 4 + 5x 2 + x – 4 Write terms in descending order by degree. Leading coefficient: 8 Terms: 4 Highest Degree: 4 Label: Quartic with 4 terms Polynomial

6/24/2016 4:17 AM6.1 - Polynomials10 Q UICK E XAMPLES 1.(x + 2) + (x + 3) 2. x + 2 – x (x 2 – 6x + 5) + (3x 2 + x – 4) 4. (2x 2 + 3x) – (3x 2 + x – 4) 5. –3x 2 + 5x x 2 + x – 8 2x x 2 – 5x + 1 –x 2 + 2x + 4 5x 3 + x 2 + x – 6

6/24/2016 4:17 AM6.1 - Polynomials11 A DD /S UBTRACT P OLYNOMIALS Keep It Simple, Students Combine like terms when adding or subtracting polynomials

6/24/2016 4:17 AM6.1 - Polynomials12 A PPLICATION E XAMPLE The cost of manufacturing a certain product can be approximated by f(x) = 3x 3 – 18x + 45, where x is the number of units of the product in hundreds. Evaluate f(0) and f(200) and describe what the values represent. f(0) represents the initial cost before manufacturing any products. f(200) represents the cost of manufacturing 20,000 units of the products. f(0) = 3(0) 3 – 18(0) + 45 = 45 f(200) = 3(200) 3 – 18(200) + 45 = 23,996,445