course Mth 158 ?????z?????assignment #005
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12:58:38 R.4.36 (was R.5.30). What is the single polynomial that is equal to 8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 )?
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RESPONSE --> First you must distribute the 8 to the first set of parentheses and 6 to the second set. (32 x^3 - 24 x^2 - 8) - (24 x^3 +48 x - 12). Since there is a negative in front of the second set of parentheses change all signs to get rid of it. 32 x^3 - 24 x^2 - 8 - 24 x^3 - 48 x +12. Then you combine like terms. 32 x ^3 - 24 x^3= 8 x^3 and -8 + 12= 4 The final answer is 8 x^3 - 24 x^2 - 48 x + 4 confidence assessment: 2
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12:59:21 ** ERRONEOUS STUDENT SOLUTION: To make this problem into a single polynomial, you can group like terms together. (8-6)+ (4x^3-4x^3) + (-3x^2) + (8x) + (-1+2). Then solve from what you just grouped...2 (-3x^2+8x+1). INSTRUCTOR CORRECTION: 8 is multiplied by the first polynomial and 6 by the second. You can't isolate them like that. Starting with 8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 ) use the Distributive Law to get 32 x^3 - 24 x^2 - 8 - 24 x^3 - 48 x + 12. Then add like terms to get 8?^3 - 24?^2 - 48? + 4 **
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RESPONSE --> That is exactly what I did. self critique assessment: 3
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13:03:38 R.4.60 (was R.5.54). What is the product (-2x - 3) ( 3 - x)?
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RESPONSE --> Using foil you first multiply -2x * 3 = -6 x Then the outer terms -2 x * -x= 2 x^2. Next the inner tems. -3 * 3= -9. And finally the last terms. -3 * -x= 3x Then you combine like terms - 6x + 3x= -3x The final answer is 2x^2 - 3x -9 confidence assessment: 2
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13:04:48 ** Many students like to use FOIL but it's much better to use the Distributive Law, which will later be applied to longer and more complicated expressions where FOIL does not help a bit. Starting with (-2x - 3) ( 3 - x) apply the Distributive Law to get -2x ( 3 - x) - 3 ( 3 - x). Then apply the Distributive Law again to get -2x(3) - 2x(-x) - 3 * 3 - 3 ( -x) and simiplify to get -6x + 2 x^2 - 9 + 3x. Add like terms to get 2 x^2 - 3 x - 9. **
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RESPONSE --> I used foil and got the same answer. I can see how the distributive law gives you the same answer, but for me its easier to use foil to solve this problem. self critique assessment: 2
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13:07:00 R.4.66 (was R.5.60). What is the product (x - 1) ( x + 1) and how did you obtain your result using a special product formula?
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RESPONSE --> This is a difference of squares. I squared the x and the 1 and got x^2 - 1 as my answer. confidence assessment: 3
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13:08:16 ** Starting with (x-1)(x+1) use the Distributive Law once to get x ( x + 1) - 1 ( x+1) then use the Distributive Law again to get x*x + x * 1 - 1 * x - 1 * 1. Simplify to get x^2 +- x - x + - 1. Add like terms to get x^2 - 1. **
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RESPONSE --> I saw that the problem was a difference of squares so I just used my knowlege of the principle to find the answer. I know that the distributive law would have given me the same answer. self critique assessment: 2
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13:12:42 R.4.84 (was R.5.78). What is (2x + 3y)^2 and how did you obtain your result using a special product formula?
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RESPONSE --> I re-wrote the problem as (2x + 3y) ( 2x + 3y). Then I distributed the 2x to everything in the second set of parentheses. 4x^2 + 6xy. I used the distributive law again and distributed 3y to the second set. 6xy + 9y^2 Then I combined 6xy + 6xy= 12xy. As my final answer I got. 4x^2 + 9y^2 + 12xy confidence assessment: 1
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13:15:26 ** The Special Product is (a + b)^2 = a^2 + 2 a b + b^2. Letting a = 2x and b = 3y we get (2x)^2 + 2 * (2x) * (3y) + (3y)^2, which we expand to get 4 x^2 + 12 x y + 9 y^2. **
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RESPONSE --> I used the distributive law when I should've used the special product. I can see how the special product works and how to use it to find the soultion. self critique assessment: 2
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13:24:45 R.4.90 (was R.5.102). Explain why the degree of the product of two polynomials equals the sum of their degrees.
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RESPONSE --> When we multiply polynomials each term of the first polynomial with will multipied by each term of the second polynomial. The exponents of each term will be added. The highest powered term of the final polynomial will have a degree equal to the sum of the degrees of the two polynomials. confidence assessment: 1
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13:26:49 ** STUDENT ANSWER AND INSTRUCTOR COMMENTS: The degree of the product of two polynomials equals the sum of their degrees because you use the law of exponenents and the ditributive property. INSTRUCOTR COMMENTS: Not bad. A more detailed explanation: The Distributive Law ensures that you will be multiplying the highest-power term in the first polynomial by the highest-power term in the second. Since the degree of each polynomial is the highest power present, and since the product of two powers gives you an exponent equal to the sum of those powers, the highest power in the product will be the sum of the degrees of the two polynomials. Since the highest power present in the product is the degree of the product, the degree of the product is the sum of the degrees of the polynomials. **
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RESPONSE --> I know that the distributive law will ensure that the highest powered terms will be multiplied together, and that the degree of this term will be equal to the sum of the degrees of the polynomials. self critique assessment: 2
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13:27:32 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> I found out when you should use FOIL and when it is appropriate to use the distributive law. confidence assessment: 3
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