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Problems of the Day Simplify each expression.

This text provides problems and solutions related to simplifying expressions, prime factorization, and factoring polynomials in algebra.

mreynolds
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Problems of the Day Simplify each expression.

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  1. Problems of the Day Simplify each expression. 1. (x + 3)(x – 9) 2. (a – 2)(a – 8) 3. (3x – 7)(4x – 4) 4. (3x – 5)(3x2 – 2x + 1) 5. Find the area of a triangle whose base is (8x + 6) and height is (x – 4). x2 – 6x – 27 a2 – 10a + 16 12x2 – 40x + 28 9x3 – 21x2 + 13x – 5 A = 4x2 – 13x – 12

  2. Algebra 1 ~ Chapter 9.1 “Factors and Greatest Common Factors”

  3. Recall that when two or more numbers are multiplied, each number is a factor of the product. • For example, the number 18 has 6 factors: 1, 2, 3, 6, 9, and 18. • (1 • 18 = 18, 2 • 9 = 18, and 3 • 6 = 18) Prime Factorization

  4. Whole numbers greater than 1 can be classified by their number of factors. • A whole number greater than 1 whose only factors are 1 and itself is called aprime # • For example: 2, 3, 5, 7, 11, 13, 17, 19, … • A whole number greater than 1 that has more than two factors is called acomposite # • For example: 4, 6, 8, 9, 10, 12, 14, 15, … Prime and Composite #s

  5. a.) 22 b.) 31 c.) 24 Factors: 1, 2, 11, 22 composite 1, 22, 2, 11 Factors: 1, 31 - prime 1, 31 1, 24, 2, 12, 3, 8, 4, 6 Example 1 – List the factors of each # and then classify as prime or composite. Factors: 1, 2, 3, 4, 6, 8, 12, 24 composite

  6. When a whole number is expressed as the product of factors that are all prime numbers, the expression is called the prime factorization of the number. • For example, the prime factorization of 18 is 2 • 3 • 3 • Those #’s are all prime and multiply together to equal 18. • The prime factorization for each number is unique. Prime Factorization

  7. A monomial is in factored form when it is expressed as the product of prime #’s and its variables. • For example, 15x2y3 in factored form is 3 • 5 • x • x • y • y • y Factored Form

  8. a.) 16 b.) 40 c.) -45 -1 45 2 20 2 8 3 15 2 10 2 4 3 5 2 5 2 2 P.F. is 23 • 5 or 2•2•2•5 P.F. is -1 • 32 • 5 or -1• 3• 3• 5 Example 2 – Find the prime factorization of each number P.F. is 24 or 2•2•2•2 A negative # is factored completely when it is expressed as the product of -1 and prime #s.

  9. Two or more #’s can have some factors in common. • The greatest (largest) of these #’s is called the Greatest Common Factor (GCF) • For example, list the factors of 24 and 18 and identify the GCF • 24: 1, 2, 3, 4, 6, 8, 12, 24 • 18: 1, 2, 3, 6, 9, 18 • The common factors are 1, 2, 3 and 6 • So the GCF is 6. Greatest Common Factor

  10. a.) 10 and 15 b.) 13 and 20 c.) 18, 45, and 63 10: 1, 2, 5, 10 The GCF is 5. 15: 1, 3, 5, 15 The GCF is 1. 13: 1, 13 20: 1, 2, 4, 5, 10, 20 Example 3 – Find the GCF of each set of #s 18: 1, 2, 3, 6, 9, 18 The GCF is 9. 45: 1, 3, 5, 9, 15, 45 63: 1, 3, 7, 9, 21, 63

  11. a.) 15a2 and 27a b.) 36x2y3 and 54xy2z 15a2: 1, 3, 5, 15, a, a The GCF is 3a 27a: 1, 3, 9, 27, a Example 4 – Find the GCF of each set of monomials 36x2y3: 1, 2, 3, 4, 6, 9, 12, 18, 36, x, x, y, y, y 54xy2z: 1, 2, 3, 6, 9, 18, 27, 54, x, y, y, z The GCF is 18xy2

  12. Ex. 5 - The area of a rectangle is 32 in2. If the length and width are both whole #s, what is the maximum perimeter of the rectangle? Remember that the area of a rectangle is L • W Possibilities (where all of the areas will be 32 in2) 1 and 32 in 2 and 16 in 4 and 8 in Perimeter = 66 in Perimeter = 36 in Perimeter = 24 in The maximum perimeter would be 66 in when the dimensions are 1 in by 32 in.

  13. Lesson Review Write the prime factorization of each number. 1. 40 2. -44 Find the GCF of each pair of numbers. 3. 16 and 48 4. 20 and 52 Find the GCF each pair of monomials. 5. 9x and 33x3 6. 45x2 and 60x3y2 23  5 -1· 22 11 16 4 3x 15x2

  14. Algebra 1 ~ Chapter 9.2 day 1 Factoring out the GCF

  15. In Chapter 8, you learned to multiply a polynomial by a monomial using the Distributive Property. • In this lesson, we will reverse that process to factor polynomials • Factoring a polynomial means to find its completely factored form • A polynomial is fully factored when it is written as a product of monomials and polynomials whose terms have no common factors other than 1. Factoring by using the Distributive Property

  16. Neither 2 nor 3x– 4 can be factored. Fully Factored 2(3x– 4) 6x– 4 can be factored. The terms have a common factor of 2. Not Fully Factored 2(6x– 4)

  17. To factor the polynomial 16c2 + 40c Step 1: Identify the GCF Step 2: Factor out (or divide out) the GCF from each of the original terms Step 3: Check your work by distributing GCF: 8c 16c2 + 40c 8c(2c + 5) Example Completely factored form! CHECK: 8c(2c + 5) 16c2 + 40c

  18. a.) 25x + 10 b.) 12y3 – 24y2 c.) 12x2 – 4x + 6 d.) 9ab3 + 27a2b – 45ab 5(5x + 2) 2(6x2– 2x + 3) 12y2(y – 2) Example 1 – Factor each polynomial 9ab(b2 + 3a – 5)

  19. a.) 24a + 30a3 b.) 14x2y3 – 21xy2 6a(4 + 5a2) c.) 12r3 – 24r2 + 20r d.) 45c2d3 – 12c2d – 3c3d2 4r(3r2– 6r + 5) 7xy2(2xy – 3) Example 2 – Factor each polynomial 3c2d(15d2– 4 – cd)

  20. If a polynomial has 4 terms, often we can factor this polynomial by “grouping”. • Example, factor 3ab + 15a + 4b + 20 (3ab + 15a) + (4b + 20) 3a(b+ 5) + 4(b + 5) (3a + 4)(b + 5) Group together terms that have common factors. Factor out GCF from each grouping. Factor by Grouping Rewrite the terms in factored form.

  21. a.) 2xy + 7x + 2y + 7 b.) 35a - 5ab + 21 - 3b (35a - 5ab) + (21 - 3b) 5a(7- b) + 3(7- b) (5a + 3)(7 - b) (2xy + 7x) + (2y + 7) x(2y+ 7) + 1(2y+ 7) (x + 1)(2y + 7) Example 3 – Factor by grouping

  22. a.) 4m2 + 4mn + 3mn + 3n2 b.) 12a2 + 3a – 8a – 2 (4m2 + 4mn) + (3mn + 3n2) 4m(m+ n) + 3n(m+ n) (4m + 3n)(m + n) (12a2 + 3a) + (-8a – 2) 3a(4a + 1) + -2(4a + 1) (3a – 2)(4a + 1) Example 4 – Factor by grouping

  23. Skills Practice 9-2 (#’s 1-18) • Practice 9-2 (#1-18) • Quiz on Sections 9-1 to 9-2 on Friday, April 15th!!!!! Assignment

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