Assigment 5

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course Mth 158

1/31/13 around 9:20AM

005. `*   4 

*   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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Your solution:

8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 )

 32x^3 - 24x^2 - 8 - 24x^3 - 48x + 12

 add/subtract like terms:

 8x^3 - 24x^2 - 48x + 4

 

 

confidence rating #$&*: 3

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Given Solution:

* *  ** 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 need to follow the order of operations.

 

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

8x^3 - 24x^2 - 48x + 4 **

 

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Self-critique (if necessary): OK

 

 

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Self-critique Rating: OK

 

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Question:

*   R.4.60 (was R.5.54). What is the product (-2x - 3) ( 3 - x)?

 

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Your solution:

 (-2x - 3) ( 3 - x)

 Using FOIL:

 -6x + 2x^2 - 9 + 3x

add/subtract like terms and put in standard form:

 2x^2 - 3x - 9

 

 

confidence rating #$&*: 3

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Given Solution:

* *  ** 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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Self-critique (if necessary): OK

 

 

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Self-critique Rating: OK

 

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Question:

*   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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Your solution:

 (x - 1) ( x + 1)

 FOIL or the distributive law:

 x^2 + x - x - 1

 simplified:

 x^2 - 1

 

 

confidence rating #$&*: 3

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Given Solution:

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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Self-critique (if necessary): OK

 

 

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Self-critique Rating: OK

 

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Question:

*   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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Your solution:

 (2x + 3y)^2 = (2x + 3y)(2x + 3y)

 FOIL or distributive law:

 4^2 + 6xy + 6xy + 9y^2

 simplified:

 4x^2 + 9y^2 + 12xy

 

 

confidence rating #$&*: 3

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Given Solution:

* *  ** The Special Product is

 

• (a + b)^2 = a^2 + 2 a b + b^2.

 

Letting a = 2x and b = 3y we substitute into the right-hand side a^2 + 2 a b + b^2 to 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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Self-critique (if necessary):

 ????? I put the terms in a different order than in the given solution. I know it wouldn't effect the solution because it doesn't matter what order you add things in, but is there a standard form for when you have a mix of monomials with exponents and monomials with more than one variable?

 

 

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Self-critique Rating: 3

@&

Either order would be appropriate to this question.

The order of my solution was dictated by the common form a^2 + 2 a b + b^2.

Your order also makes sense, with the highest powers of each variable represented first.

*@

 

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Question:

*   R.4.105 \ 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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Your solution:

 I know this is referring to problems like this:

 x^2 * x^4 = x^6

 I'm not really sure how to explain it. When you raise something to a power (2^3) you are multiplying that number by itself equal to the exponent (2^3 = 2*2*2 = 8). When you multiply polynomials raised to whatever power (2x^3 * 3x^2) you add their degrees to simplify the process (2*3 * x ^3+^2 = 6x^5).

 

 

confidence rating #$&*: 1

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Given Solution:

* *  ** 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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Self-critique (if necessary):

 I'm not sure I fully understand this. I know how to do that, but I'm still not sure why.

 

 

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Self-critique Rating: 1

@&

Your explanation is good. It appears to me that you understand this.

*@

 

*   Add comments on any surprises or insights you experienced as a result of this assignment."

Self-critique (if necessary):

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Self-critique rating:

 

*   Add comments on any surprises or insights you experienced as a result of this assignment."

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#*&!

&#This looks good. See my notes. Let me know if you have any questions. &#