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3.2.3 Multiplying Polynomials and the Distributive Property Name:_____________________________ I can: ____________________________________________________________________________________. Double Distributive Property or FOIL - can only be used to multiply two binomials Example: Multiply (3x – 4) ( x + 2). Example: Multiply (–5x + 7) (–3x – 6) Generic Rectangle - can be used to multiply any polynomials (factor 1) Example: Multiply (2x – 4) (–3x2 + 5x – 6) ____________________________________ (factor 2) Example: Multiply (–x + 3)2 ____________________________________ Example: Multiply 3x(x2 + 7x – 5) ____________________________________ Practice Use the Distributive Property(FOIL) to multiply the following; 1. (x + 3)(x + 8) 2. (y – 4)(y + 7) 3. (3y) (x – y) 4. (2x)(x – 1) 5. (2r – 5)(r + 3) 6. (3x + 1)(5x – 3) 7. (x – 3)2 8. (5t – 1)(3t – 2) 9. (a + 11)(11a + 1) Use Generic Rectangles to multiply the following; 1. (4x – 7) (2x – 9) 2. (4x + 3) (7x – 5) 3. (–3x + 5)2 4. (x + 2)(4x – 3) 5. (4x – 7)2 6. (3x – 5) (2x2 – 4x + 7) 7. (4x + 3) (3x2 – 5x – 6) 8. (2x2 – 5x) (9x2 + 6x + 4) 9. (2x2) (3x2 + 5x - 4) The Generic Rectangle Challenge - Find the missing terms and write area in factored form = area as sum 1. 2. -3 ____ x 2x2 _____ _____ 3. +5 ____________________ = ____________________ -2 -5x2 _____ 3x3 x2 +1 5. -16x +4 _____ ____________________ = ____________________ _____ +5 ____________________ = ____________________ 6. +7 ____ 5x _____ 4. ____ x 27x3 9x2 +1 ____________________ = ____________________ -18x -2 -15x2 _____ _____ +1 ____________________ = ____________________ +16x +16 12x4 4x2 _____ +12 ____________________ = ____________________ 3-57 CLOSED SETS Do you remember the subsets of REAL numbers? Whole numbers (positive integers and zero) are said to be a closed set under addition: If you add two whole numbers, you always get a whole number. 15 + 23 = 38 a whole number (This is an example that proves the point) On the other hand, whole numbers are not a closed set under subtraction: If you subtract two whole numbers you may not get a whole number: 15 – 23 = –8 an integer (This is a counter-example that disproves the point.) Answer the following questions; 1. Are whole numbers a closed set under multiplication? Use examples and counter-examples to explain. 2. Are whole numbers a closed set under division? Use examples and counter-examples to explain. Now, use the same process to examine integers and closed sets. 1. Are integers a closed set under addition? Use examples and counter-examples to explain. 2. Are integers a closed set under subtraction? Use examples and counter-examples to explain. 3. Are integers a closed set under multiplication? Use examples and counter-examples to explain. 4. Are integers a closed set under division? Use examples and counter-examples to explain.