Warm-Up #23 (Monday, 10/26) Simplify (x+2)(3x – 4) 2. Simplify (x+2)( x 2 −3x−1) 3. Factor 16x 2 −25

Homework (Monday, 10/26) lesson 2.06_page 98 #11, 12, 13

Lesson 2.06 Anatomy of a Polynomial

Monomial A monomial is the product of a number, the coefficient, and one or more variables raised to nonnegative integer powers. If a monomial has more than one variable, the degree is the sum of the exponents of each variable Example: 3𝑥 2 6𝑛 3 𝑦 4

Polynomial A polynomial is a monomial or a sum of two or more monomials Each term contains only variables with whole number exponents and real number coefficients Examples: 3x + 7 7𝑥 2 −5𝑥+3 2 𝑥 4 𝑦 3 + 3𝑥𝑦 2

Standard Form A polynomial is in standard form when its terms are written in descending order of exponents from left to right Examples: 2x + 7 14𝑐 3 −5𝑐+8 4𝑎𝑏 2 −3𝑎𝑏+2

𝟑𝒙 𝟑 − 𝟓𝒙 𝟐 −𝟐𝒙+𝟏 Parts of a polynomial Cubic term Linear term Quadratic term Constant Leading coefficient

Degree of the Term The exponent of the variable in the term determines the degree of the term Example: The highest degree of 12𝑑 5 is 5 or fifth degree What is the highest degree of 14𝑐 3 ? Since the exponent is 3, then the term is of degree three or cubic. What is the highest degree of 14𝑐 3 - 7c + 3? This is third degree, or cubic, polynomial since the highest exponent is 3.

You Try: What is the highest degree and leading coefficient for each polynomial 14𝑐 3 + 2𝑐 2 −𝑐+1 3𝑐+ 2𝑐 2 −1 𝑐 5 + 2𝑐 1 −3𝑐+6 4𝑐 3 + 2𝑐 4 −2𝑐 d: c:

Classifying Polynomials by Number of Terms Example Number of Terms Name 14c 1 Monomial 2c – 7 2 Binomial 14𝑐 2 +8𝑐 −5 3 Trinomial 14𝑐 3 + 2𝑐 2 −𝑐+1 4 polynomial ANYTHING MORE THAN A MONOMIAL IS CONSIDER A POLYNOMIAL

Classifying Polynomials by Degree Example Degree Name 14 Constant 2x – 7 1 Linear 14𝑥 2 +8𝑥 −5 2 Quadratic 14𝑥 3 +16 3 Cubic 14𝑥 4 +7𝑥−2 4 Quartic −4𝑥 5 − 𝑥 3 5 Quintic

Recognize Degree There is no special name after the 5th degree. You only write it as 6th degree, 7th degree, etc. The degree determines how many possible solutions/x-interceptions the function can have.

Classifying the Polynomial Write each polynomial in standard form and classify it by degree and number of terms 5𝑥 3 +4𝑥+ 2𝑥 2 +1 7𝑥 5 + 3𝑥 3 −2𝑥+4

Find a polynomial (an example) that satisfy each condition Find a polynomial (an example) that satisfy each condition. If it is impossible to satisfy the condition, explain why. A cubic polynomial with three terms, all of different degrees A linear polynomial with two terms, both of different degrees A linear polynomial with three terms, all of different degrees

Is each result always a polynomial? Explain Sum of two polynomials Product of two polynomials Difference of two polynomials Quotient of two polynomials