R4

course Mth 158

6/6 8

If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution:

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

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:

I will group like terms and simplify.

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

(8 – 6=(2)) -3x^2 + 8x +1

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

confidence rating #$&* 2

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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):

Of course! I forgot the most important rule. PEMDAS.

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Self-critique rating #$&*

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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)

I will find the solution step by step without foil.

-2x(3)= -6x, -2x(x)= -3x, -3(3)=-9, -3(-x)=3x

-6x + 2x squared -9 -9x

combine like terms

2x squared -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 simplify 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 #$&*

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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)

x squared + x –x - 1

combine like terms

x squared - 1

I didn’t take notes on the “special formulas” in the book because I try not to mess with something that already works. However, if you feel they are important to understanding the concepts for this course I will reluctantly add them to my notes.

I generally encourage basic principles, and you should certainly see how special formulas come from the basic principles . However those formulas appear so often and come in so handy that they are worth remembering.

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):

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Self-critique rating #$&*

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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)

4x squared + 6xy +6xy +9 y squared

combine like terms

4x squared + 9y squared +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):OK

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Self-critique rating #$&*

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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:

The degree of the product of two polynomials equals their sum degreases due to the laws of distribution.

confidence rating #$&* 2

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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 understand this more clearly now.

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Self-critique rating #$&*

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

"

&#Very good work. Let me know if you have questions. &#

#$&*

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