Chapter 6 Exponents and Polynomials What You’ll Learn: Exponents Basic Operations of Nomials

In Class Assignment Page 389 # 1-32

Integer Exponents Zero Exponents – Any nonzero number raised to the zero power is 1 Negative Exponents – A nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent.

Integer Exponents Power Value Power Value

Ex. 1) Zero and Negative Exponents Simplify. A.2 -3 B.(-3) -4 C.-3 4

Ex.2 Evaluating Expressions Evaluate. A.x -1 for x = -2 B.a 0 b -3 for a=8, b = -2

Ex. 3 Simplify Expressions Simplify. A. 3y -2 B.-4 k -4 C. x -3 a 0 y 5

Homework Practice 6-1 (page395) – #’s 2-22 even

Rational Exponents The radical symbol is used to identify roots The index is the small number to the left that tells which root to take Roots of 2,3,4,5,ect…

Writing Roots with Exponents b = b k

Ex. 1) Writing Roots with Exponents Simplify (w/ and w/out calc) 125 1/3

Ex.2 Simplify Expressions x 9 y 3 3

Homework Practice 6-2 (page401) – #’s 2-9;23-30

Polynomials Monomial (Term) is a number, a variable, or a product of numbers and variables with whole number exponents.  Ex. Polynomial is a monomial or a sum or difference of monomials  Ex.

Finding the Degree of Nomials Monomial – Add the exponents of variables Ex. -2a 2 b 4 >>> Polynomial is a monomial or a sum or difference of monomials – Find the use the degree of the term with the greatest degree  Ex. 4x – 18x 5

Finding the Degree of Nomials Find the degree a.4x b.2c 3 c.X 3 y 2 + x 2 y 3 – x 4 + 2

Writing Polynomials in Standard Form Standard form of a polynomial – Polynomials are written with terms arranged in descending order (greatest degree downward)

Polynomials can also be classified based on their degree or by how many terms it contains By degree 0 - Constant 1- Linear 2- Quadratic 3- Cubic 4- Quartic 5-Quintic 6 or more- 6 th, 7 th, 8 th degree….. By terms 1 – Monomial 2 – Biomial 3 – Trinomial 4 or more – Polynomial

Classifying Polynomials Classify o 5x – 6 o y 2 + y + 4 o 6x 5 + 9x 4 – x + 3

Homework Practice 6-3 (page 409) #’s 1-3;4-24 (even)

Algebraic Expression In an algebraic expression, a positive or negative sign is part of the term that follows it: the term owns the sign that comes before it. – An additions sign is understood in front of a negative sign

Like Terms Like Terms have the same variable or variables raised to the same power. – In other word in order to be like, Terms can have different first names (numbers), but must have the same last name (letter & power)

Examples of Like Terms 4x and 9x. Both have x 7xy and 8 xy. Both have xy 2y and 8y. Both have y Can you think of other like Terms 2 2 2

Non Example of Like Terms 4 and 6y. Do not have common factor 3x and 3y. Have common factor by not variable 5y and 6y. Do not have the same power Can you think of other examples??? 2

Combining (add/subtract) Like Terms Simplify: 4x + 6y – 3x – 4y Group like terms (x & y terms) [the sign travels with the terms] (4x + - 3x) + (6y + - 4y) Combine like terms – x + 2y

Adding Polynomials Add 3x + 4xy – 2y + 3 and x y + 3y - 4 – Group like terms – Combine like terms You can also align similar terms vertically and then add

Practice Adding Polynomials Alg Bk. p. 149 Simplify Horizontally 3xy + 4x – 2y + 3 and x y + 3y - 4 Simplify Vertically 3x y + 4x – 2y + 3 x y + 3y - 4 _________________

Subtract Polynomials Subtract –a - 5ab + 4b - 2 from 3a – 2ab – 2b Add the opposite of the second polynomial – Change sign of terms Group like terms Combine like terms 2 222

Simplify : ( –a - 5ab + 4b) – (3a – 2ab – 2b) (3x 2 - 2x + 8 ) - (x 2 - 4) (4b 5 + 8b) + (3b 5 + 6b - 7b 5 + b)

Homework Practice 6-4 (page 417) o Day 1 o #’s 1-14 o Day 2 o #’s 16-32

Exponential Notation An exponent is a number that represents how many times the base is used as a factor. For example, the number 8 with an exponent of 4 is equal to 8 x 8 x 8 x 8. Base Exponent

Multiplying Monomials Rule of Exponents for products of powers – To multiply two powers with the same base, you add the exponents (x ) (x ) = x (2x ) (4x ) = 8x

Multiply Monomials (x ) (x ) (y ) (y) (y ) 2s(5s) (4y z)(2yz ) (5x y)(3x y ) (-3s) (7s )

Simplify Products of Monomials (3x y )(-2x y) + (8x y ) (x y )

Multiplying Polynomials By using the distributive property and the rules of exponents, any polynomial can be multiplied Two methods for doing so: – Horizontal method – Vertical method

Multiply x (x+3) Horizontal Method – x (x + 3) x + 3x Vertical Method x + 3 x ____________ x + 3x 2 2

Multiply -2x(4x - 3x + 5) Horizontal MethodVertical Method

Multiply 5xy (3x - 4xy + y ) Horizontal MethodVertical Method 22 2

Multiply (3x – 2) (2x - 5x – 4) 2x -5x - 4 3x – 2 _____________ Multiply by the 3x first Multiply by the -2 Add/combine like terms 2 2

Answer: 6x - 19x -22x + 8 It is helpful to rearrange the terms in either ascending or descending order – Descending order x + 2x - 4x + 2 – Ascending order 2 – 4x + 2x + x

Multiplying Binomials When multiplying binomials the product results in a trinomial – (a + b) (c + d) = ac + ad + bc + bd In to multiply we must use the distributive property We call this Method of multiplication FOIL

FOIL (a + b ) (c + d) F – Firsts – (a + b ) (c + d) = ac O – Outers – (a + b ) (c + d) = ad I – Inners – (a + b ) (c + d) = bc L – Lasts – (a + b ) (c + d) = bd

Write the product of (2x + 5) (3x- 4) FOIL – Firsts: (2x)(3x) = 6x – Outers: (2x)(-4) = -8x – Inners: (5)(3x) = 15x – Lasts: (5)(-4) = -20 6x – 8x +15x – 20 6x + 7x

Practice Foil (x+1)(x+8) (y+2)(y+5) (t-5)(t-3) (u-2)(u-1) (s-9)(s+9) (8k-1)(k+3) (2n+4) 2

Homework Practice 6-5 (page 427) o Day 1 o #’s 2-24 even o Day 2 o #’s 1-23 odd

Special Product Binomials Perfect-Square Trinomial - A trinomial that is the result of squaring a binomial  (a + b ) 2  (a + b) (a – b)

Ex 1) Find the Product (a + b ) 2 Multiply o (x + 4) 2 o (3x + 2y) 2 o (4 + s 2 ) 2

Ex 2) Find the Product (a - b ) 2 Multiply o (x - 5) 2 o (6a - 1) 2 o (3 – x 2 ) 2

Ex 2) Find the Product (a + b ) (a – b ) Multiply o (x + 6 ) (x – 6 ) o (x 2 + 2y ) (x 2 – 2y )

Homework Pr 6-6 (p. 437) o Day 1 # even o Day 2 # odd

Chapter Review Text p – 445 – Selected problems