Honors Advanced Algebra Lesson 2-1

Warm-up Simplify the following by combining like terms. 1. 2xyz + 4x + 5y + 2x – 3z – 2y – xyz 2. 3x 2 + 4x 3 – 2x + x 3 – x 2 + 2x Challenge: 5(x + y 2 ) – 3x + 6y 2 – 18x ÷ 3

Vocabulary A Polynomial is an expression composed of terms or the sum of terms. Polynomials are named based upon the number of terms that compose them. A Monomial – one term A Binomial – two terms A Trinomial – three terms In order to define a polynomial, make sure that the expression is simplified first.

Examples Monomial: 8.06t x 2 Binomial: 6x + 4 x 2 – 3 3x 2 + x Trinomial: x 2 + 6x x 3 – 2x 2 +5

Degree of Monomials The degree of a monomial is the sum of the exponents of its variables. If the monomial is a nonzero constant, the degree is 0. The constant 0 has no degree.

Degree of a Monomial BACK Exponent of x is = 7 Nonzero constant

Degree of a Polynomial The degree of a polynomial is the highest degree of any of its terms after it has been simplified. Ex: 6x 3 + 4x The degree of this polynomial is 3 because that is the highest exponent after the polynomial has been simplified.

Let’s Practice Name the polynomial and tell me the degree of the following: 1. 5x x 5 + 4x 3 – x 3. 9 – x 2 Challenge: 1. 5x 2 y – 3xy x 3 - 2x 7 + 1/2x 3

Names Based on Degree Linear Quadratic Cubic Quartic

Descending Order/Standard Form It is standard form to write polynomials in descending order by degree of each term. Degree 4, 3, 2, 1,

Let’s Practice Write in descending order of “x” and tell me the name based on degree. 1. x 2 - 3x x + 5x 4 – x 2 – x x Challenge 3x 2 + x 5 + x 3 - x 2

Adding Polynomials To add or subtract, add/subtract the coefficients of like terms! Add 3x 3 +2x 2 -x-7 and x 3 -10x x 3 + 2x 2 – x – 7 + x 3 – 10x Line up like terms 4x 3 – 8x 2 – x + 1

Example Add 5x 3 +8x 2 -2x-4 and -10x 2 +6x 3 +3x-7. 5x 3 + 8x 2 – 2x – 4 + 6x 3 –10x 2 + 3x - 7 Line up like terms 11x 3 – 2x 2 + x - 11

Let’s Practice Add the following polynomials together (9x 3 – 2x + 1) + (5x x -4)

Subtracting Polynomials To subtract polynomials, we simply subtract the coefficients of the 2 nd polynomial from the coefficients of the first. Subtract 3x 3 +2x 2 -x-7 and x 3 -10x x 3 + 2x 2 – x – 7 - x x Line up like terms 2x 3 +12x 2 – x - 15 Switch the signs of The right expression

Let’s Practice Find the difference of the following polynomials. 1. (5x 2 +x – 7 ) – (3x 2 + 6x + 1) 2. (x 4 + 2x 3 + 8) – (2x 4 – 9)

Homework Pg , #19-31, 41, 42