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 )?
Your solution:
Confidence Assessment:
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
**
Self-critique (if necessary):
Self-critique Rating:
Question:
* R.4.60 (was R.5.54). What is the product (-2x - 3) ( 3 - x)?
Your solution:
Confidence Assessment:
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. **
Self-critique (if necessary):
Self-critique Rating:
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?
Your solution:
Confidence Assessment:
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. **
Self-critique (if necessary):
Self-critique Rating:
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?
Your solution:
Confidence Assessment:
Given Solution:
* * ** The Special Product is
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.
**
STUDENT QUESTION:
Uh-oh, apparently that wasn’t the correct
way to solve it. Distributive Law doesn’t apply here - if that’s the case, the
answer I have is incorrect and I need more clarification on problems like these.
<h3>@&
The Distributive Law does apply, but this time we're using the special formula,
which comes from the distributive law and applies to the square of any binomial.
(a+b)^2 = (a + b) (a + b)
= a ( a + b) + b ( a + b)
= a^2 + ab + ba + b^2
= a^2 + 2 a b + b^2.
That's how the special formula is derived.
That special formula applies to this problem, as shown in the given solution.
*@</h3>
<h3>@&
It would have been possible to do this using just the distributive law:
(2x + 3y)^2 = (2x + 3y) ( 2x + 3y)
= 2x ( 2x + 3y) + 3y ( 2x + 3y)
= 4x^2 + 6 x y + 6 y x + 9 y^2
= 4 x^2 + 12 x y + 9 y^2.
The special formula, though, allows you to get to the correct result much
quicker.
Since we often need to square binomials, the special formula is frequently used,
and as long as you know how to square a binomial using the distributive law (as
just shown in this note) you can save time by using the special formula.
*@</h3>
Self-critique (if necessary):
Self-critique Rating:
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.
Your solution:
Confidence Assessment:
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. **
Self-critique (if necessary):
Self-critique Rating:
* Add comments on any surprises or insights
you experienced as a result of this assignment.