Polynomials

In This Unit… Review simplifying polynomials, distributive property & exponents Classifying Polynomials Area & Perimeter with Algetiles Factoring in Algebra Multiplying & Dividing Monomials Multiplying Polynomials & Monomials Factoring Polynomials Dividing Polynomials Multiplying Two Binomials Factoring Trinomials

A monomial is a number, a variable, or a product of numbers and variables. A polynomial is a monomial or a sum of monomials. The exponents of the variables of a polynomial must be positive. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. The degree of a monomial is the sum of the exponents of its variables. To find the degree of a polynomial, you must find the degree of each term. The greatest degree of any term is the degree of the polynomial. The terms of a polynomial are usually arranged so that the powers of one variable are in ascending or descending order.

Classifying Polynomials A monomial is an expression with a single term. It is a real number, a variable, or the product of real numbers and variables. Example: 4, 3x2, and 15xy3 are all monomials

Classifying Polynomials A binomial is an expression with two terms. It is a real number, a variable, or the product of real numbers and variables. Example: 3x + 9

Classifying Polynomials A trinomial is an expression with three terms. It is a real number, a variable, or the product of real numbers and variables. Example: x2 + 3x + 9 Now you try to Classify Each

POLYNOMIAL Monomial Binomial Trinomial 2x + 9 x 3 10x2 + 2x + 9 2(x + 4) 3x + 4 6x - 8 -9x 3x2 + 3xy + 9x 10 2x x2 + 3xy + 9xyz

Algebra Tiles & Area x x2 - tile x 1 x-tile x 1-tile 1 1-tile 1

Draw algebra tiles to represent the polynomial 3x2 – 2x + 5 Recall This Algebraic Expression has 3 Terms: 3x2 3 is the coefficient, x2 is the variable –2x -2 is the coefficient, x is the variable 5 5 is the constant term

What is the area of a rectangle? AREA = Length x Width How do you find the perimeter of a rectangle? ADD up all of the sides

Length = 5 This rectangle has the following properties: Width = x We can combine algebra tiles to form a rectangle. We can then write the area and the perimeter of the rectangle as a polynomial. This rectangle has the following properties: Length = 5 Width = x Perimeter is x + 5 + x + 5 = 2x + 10 Area = LW = 5 x x = 5x x 5

Determine the Area & Perimeter of the following Rectangles x x x x This rectangle has the following properties: Length = 3x Width = x Perimeter = x + 3x + x + 3x = 8x Area = (3x) * (x) = 3x2

1.) 3.) 2.) Length ________ Width ________ Area ______________________ Perimeter __________________ Width ________ Length ________ 3.) 1.) 2.)

2.) Length x Width 2 Perimeter x + x + 2 + 2 = 2x + 4 Area (4) * (x) = 4x Perimeter x + 4 + x + 4 = 2x + 8 Width x Length 4 3) Area (2x) * (2x) = 4x2 Perimeter 2x + 2x + 2x +2x = 8x Width 2x Length 2x Area (2) x (x) = 2x Perimeter x + x + 2 + 2 = 2x + 4 Width 2 Length x 1) 2.)

Multiplying Monomials RECALL : Multiplying Powers: When multiplying powers with the same base we add the exponents Example: x2 * x2 = x4 Dividing Powers: When dividing powers with the same base we subtract the exponents Example: x3 ÷ x1 = x2 Power of a Power: Example:

Multiplying Monomials (3x2)(5x3) = (3 * x * x) (5 * x * x * x) = (3) (5) (x*x*x*x*x) = 15x5

With Algetiles x * x = x2 (2)(5x) = 10x

So 3 x 2 x 2 are prime factors of 12 Prime Factor Review A prime factor is a whole number with exactly TWO factors, itself and 1 A composite number has more than two factors 12 12 FACTOR TREES 4 3 2 6 2 3 So 3 x 2 x 2 are prime factors of 12

Practice Exercises: Express each number as a product of its prime factors: 30 36 25 42 75 100 121 150

Practice Solutions: 2 x 3 x 5 2 x 2 x 3 x 3 5 x 5 2 x 3 x 7 3 x 5 x 5

We can factor in algebra too 3x2 = 3 * x * x 5x = 5 * x 2x4 = 2 * x * x * x * x 2x2y2 = 2 * x * x * y * y Let’s Try: a)4x3 b) –x2 c)2x6 d) 9x2y e) -6a2b2

We can factor in algebra too a)4x3 = 4 * (x * x * x ) b) –x2 = (-1) * (x * x) c)2x6 = (2) * (x * x * x * x * x * x) d) 9x2y = (9 * 2) * (x) * (y) e) -6a2b2 = (-6) * (a * a) * (b * b)

Greatest Common Factor The greatest of the factors of two or more numbers is called the greatest common factor (GCF). Two numbers whose GCF is 1 are relatively prime.

List the common prime factors in each list: 2, 3. Finding the GCF To find the GCF of 126 and 60. = 2 x 3 x 3 x 7 60 = 2 x 2 x 3 x 5 List the common prime factors in each list: 2, 3. The GCF of 126 and 60 is 2 x 3 or 6.

List the common prime factors in each list: Finding the GCF Find the GCF of 140y2 and 84y3 140y2 = 2 * 2 * 5 * 7 * y * y 84y3 = 2 * 2 * 3 * 7 * y * y * y List the common prime factors in each list: 2, 2, 7, y, y The GCF is 2 * 2 * 7 * y * y = 28y2

Finding the GCF Try These Together What is the GCF of 14 and 20? 2. What is the GCF of 21x4 and 9x3? HINT: Find the prime factorization of the numbers and then find the product of their common factors.

Finding the GCF What is the GCF of 14 and 20? Factors of 14 = 2, 7 Therefore the GCF is 2 2. What is the GCF of 21x4 and 9x3? Factors of 21x4 = 3 * 7 * x * x * x * x Factors of 9x3 = 3 * 3 * x * x * x Therefore the GCF is 3* x * x * x = 3x3