Algebra I : Semester II : Polynomials

Sections:

Introduction  |   Section 1  |   Section 2  |   Section 3  |  Section 4 | Section 5  |  Section 6

Section Four

Part 1  |   Part 2   |  Part 3  |  Part 4

Algebra 1: Section 4: Polynomials Prime Factorization and GCF (Greatest Common Factor) Did you ever watch the film Contact?  Jodie Foster, who plays an astronomer in the movie, believes that she is getting a message from aliens when the sounds in the message count out the first 100 prime numbers.  Primes are considered important because they are the building blocks of all of the rest of the whole numbers. To see this, you can use prime factorization, the number represented as a product of primes.       Steps for Finding the Prime Factorization of a Number There are two methods to do this. Method 1 Write down the number you wish factor. Let’s use 36 for this example. Write any two numbers that multiply to give a product of 36. If the number is prime, write it down again; if it is composite, write it as a product of two numbers. Continue until all numbers are primes. Write the number as a product of its primes. Method 2 Write the number in an upside down division symbol. Divide the number by any prime that will divide into it evenly. You may want to review the first eight primes in your list: 2, 3, 5, 7, 11, 13, 17, 19. Divide the number again by any prime number that will divide into it evenly. Keep going until the last number you wrote down is a prime number. Write the number as a product of primes. It really does not matter which method you use; they both should give the same result. Example 1: Prime Factorization Find the prime factorization of 280. 280 = 23∙5∙7 Example 2: Prime Factorization of a Monomial Factor each monomial completely. 24x3y2 24x3y2 = 2·2·2·3·x·x·x·y·y in completely factored form. -20xy2 -20xy2 = -1·2·2·5·x·y·y in completely factored form. The greatest common factor or GCF also uses prime factorization.   Greatest Common Factor (GCF) The GCF of two or more integers is the product of the prime factors common to the integers. The GCF of two or more monomials is the product of their common factors, including the variables, from factored form. If the monomials have a GCF of one, then they are said to be relatively prime. Example 3:  Finding the GCF Find the GCF of each of the following sets. 64 and 24 GCF = 2∙2∙2 = 8 15 and 60 GCF = 3∙5 = 15 84xy3 and 36x2y5 GCF = 2∙2∙3∙x∙y∙y∙y = 12xy3 Example 4: Problem Solving The area of a rectangle is 12 square feet. Its length and width are whole numbers.  What are the possible dimensions of the rectangle? To find the possible measures of the length and width, remember the area formula, A = LW. In this case, 12 = LW and the possible solutions include: (1)(12) = 12 (2)(6) = 12 (3)(4) = 12. So the possible dimensions include: 1’ x 12’ 2’ x 6’ 3’ x 4’ What is its maximum perimeter? The formula to find the perimeter of a rectangle is P = 2L + 2W. Find the solution by substituting the pairs in the formula. 2(1) + 2(12) = 26 2(2) + 2(6) = 16 2(3) + 2(4) = 14 Therefore the maximum perimeter is 26 feet.   Practice List the factors of each and classify it as prime or composite. 31 {1, 31} Prime 1 {1} Neither prime nor composite 48 {1, 2, 3, 4, 6, 8, 12, 16, 24, 48} Composite 20 {1, 2, 4 ,5, 10, 20} Composite Find the prime factorization of each. 48 2·2·2·2·3 = 24·3 -30 -1·2·3·5 150 2·3·5·5 = 2·3·52 Factor each monomial completely. 18x2yz2 2·3·3·x·x·y·z·z 54ab2 2·3·3·3·a·b·b Find the GCF of each set of monomials.  12xy and 26x2    2x  30x3y2 and 50x2yz    10x2y  15a2, 35b2, and 70ab Five is the only factor common to all 3 monomials. Solve the problem.  A rectangle has an area of 20 square centimeters (cm). The length and width are each whole numbers. What are its possible dimensions? What is the maximum perimeter from these dimensions? Possible dimensions are 1cm x 20cm, 2cm x 10cm, 4cm x 5cm.  Maximum perimeter is 2(1) x 2(20) = 42 cm. Now go on to the next part

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