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CH. 10 Factoring 10.1 Factors Factors Recall, the two numbers that are multiplied together are called the factors of the product 3 x 4 = 12 Prime Numbers Numbers that have exactly two factors, 1 and itself Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19 Composite Numbers Numbers that have more than two factors Composite numbers less then 20: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 Neither 0 and 1 are neither prime nor composite!! Example Find the factors of each number. Then classify each number as prime or composite. 47 35 25 23 Prime Factorization When a number is expressed as a product of prime factors Use a factor tree to find the prime factorization 12 Example Factor each monomial (prime factorization). 16b2c2 9c3d -15xy2 Greatest Common Factor Two numbers may have common prime factors The product of these prime factors is called the greatest common factor 6 is the GCF of 36 and 42 Examples Find the GCF of each set of numbers or monomials. 12, 8 20, and 24 and 9 Examples Find the GCF of each set of number or monomials. 21ab2 and 9a2b 24ab2c and 60a2bc Example The area of a rectangle is 24 square inches. Find the length and width so that the rectangle has the least perimeter. Assume that the length and width are both whole numbers. Example The area of a rectangle is 18 square inches. Find the length and width so that the rectangle has the least perimeter. Assume that the length and width are both whole numbers. Assignment 1st Assignment: P424: 1, 5 – 21 2nd Assignment: P424: 22 – 56 even, 58 – 68 10.2 Factoring Using the Distributive Property Factoring When you know the product and are asked to find the factors To factor means to write a polynomial as a product of monomials and smaller polynomials Factoring a Polynomial Find the GCF Write as a product of GCF and remaining factors Use the distributive property Example 2 8y + 10y Find the GCF Write as a product of GCF and remaining factors Use distributive property Example Factor each polynomial. 24y + 18y2 18fg – 21gh2 Example Factor each polynomial. 30x2 + 12x 15ab2 – 25abc Example Factor each polynomial. 5a + 20ab + 10a2 17de – 15f Example Factor each polynomial. 16a2b 20rs2 + 10ab2 – 15r2s + 5rs Finding a missing factor If you know one of the factors, you can find the other one by division To divide a polynomial by a monomial, divide each term of the polynomial by the monomial Example Divide (24a2 – 20a) by 4a. Example Divide (9b2 – 15) by 3. Example Divide (10x2y2 + 5xy) by 5xy. Example The diagram shows a walkway that is 2 meters wide surrounding a rectangular planter. Write an expression in factored form that represents the area of the walkway. Example A stone walkway is to be built around a square planter that contains a shade tree. If the walkway is 2 meters wide, write an expression in factored form that represents the area of the walkway. Assignment 1st Assignment: P431: 2, 4 – 18 2nd Assignment: P432: 20 – 46 even, 47 – 57 P712: 10-2: 2 – 30 even 10.3 Factoring Trinomials: x2 + bx + c Factoring Trinomials One way is to factor a trinomial into two binomials (x + 3)(x + 5) = x2 + 8x + 15 Example x2 + x – 12 x2 – 9x + 12 Example x2 + 3x +2 a2 + a + 3 Example b2 + 4b + 4 y2 – 7y + 12 Example n2 – 5n – 14 m2 – m + 1 Factoring Trinomials If the trinomial has a leading coefficient, always look for a GCF Example 4x2 – 8x – 60 3y2 – 9y – 54 Example 5m2 +45m + 100 2x2 – 20x – 22 Example Sahej is planning a rectangular garden in which the width will be 2 feet less than the length. He will put a composting box inside the garden that measures 2 feet by 4 feet. How many square feet are now left for planting? Express the answer in factored form. Example Tammy is planning a rectangular garden in which the width will be 4 feet less than its length. She has decided to put a birdbath within the garden, occupying a space 3 feet by 4 feet. How many square feet are now left for planting? Express the answer in factored form. Assignment 1st Assignment: P438: 3 – 18 2nd Assignment: P438: 20 – 46 even, 49, 50, 53 - 62 10.4 Factoring Trinomials: ax2 + bx + c Review Factoring Trinomials with a Leading Coefficient If can’t factor out a GCF Use reverse FOIL to factor into two binomials Example 2x2 – 7x + 3 Example 2x2 – 9x + 4 Example 3y2 + 7y – 6 Example 2x2 + 3x + 1 Example 5y2 + 2y – 3 Example 3z2 – 8z + 4 Example 4x2 – 4x – 15 Example 6x2 + 17x + 5 Example 4x2 – 8x – 5 Example The volume of a rectangular shipping crate is 2x3 – 4x2 – 30x. Find possible dimensions of the crate. Example The volume of a rectangular shipping crate is 6x3 – 15x2 – 36x. Find possible dimensions for the crate. Assignment 1st Assignment P443: 3 – 12 2nd Assignment P443: 14 – 40 even, 41 – 43, 45 – 49, 52 10.5 Special Factors Perfect Square Trinomials The square of (x + 3) is the sum of The square of the first term of the binomial The square of the last term of the binomial Twice the product of the terms of the binomial Perfect Square Trinomials Example Determine whether each trinomial is a perfect square trinomial. If so, factor it. x2 + 14x + 49 9a2 + 16a + 4 Example Determine whether each trinomial is a perfect square trinomial. If so, factor it. 16b2 a2 + 24b + 9 + 2a + 1 Example Determine whether each trinomial is a perfect square trinomial. If so, factor it. 16x2 + 20x + 25 49x2 – 14x + 1 Example The area of a square is d2 – 16d + 64. Find the perimeter. Example The area of a square is x2 + 18x + 81. Find the perimeter. Factoring a Difference of Squares Factoring a Difference of Two Squares Example Determine whether each binomial is a difference of squares. If so, factor it. d2 – 81 f2 + 64 Example Determine whether each binomial is a difference of squares. If so, factor it. 4m2 – 144 121 – p2 Example Determine whether each binomial is a difference of squares. If so, factor it. 25x3 4a2 – 100x + 49 Summary Assignment 1st Assignment P448: 3 – 12 2nd Assignment P448: 14 – 52 even, 54 – 62 Extra Practice P711: 10-1: 1, 9, 11, 20 – 25 10-2: 3 – 30 x 3s 10-3: 2 – 14 even 10-4: 2 – 16 even