YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
Multiplication, Addition and Subtraction of Polynomials
Note: The column method for multiplication also works when the coefficients are negative and the variable b (or x) is allowed to take real values. But in the first case we assume all coefficients, the variable b (or x) below is positive.
A. Multiplication
The following animated example shows how to calculate or expand the product of two polynomials. In it, assume x is a positive number or length.
If this animation is too fast for your liking, write the details on paper, one detail at a time and one detail after another.
B. More on Multiplication
Here we use the letter b instead of the letter x to denote a positive number or length.
Problem: Use rectangles to expand the product
P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3)
where b is a whole number
Solution: Form intermediate size rectangles - sides here and below are not drawn to scale.
| 10b3 | 4b2 | 6b | 8 | |
| 50 | ||||
| 6b | ||||
| 4b2 |
and compute the number of subrectangles in each. The latter operation first gives
| 10b3 | 4b2 | 6b | 8 | |
| 50 | 50 x 10b3 | 50 x 4b2 | 50 x 6b | 50 x 8 |
| 6b | 6b x 10b3 | 6b x 4b2 | 6b x 6b | 6b x 8 |
| 4b2 | 4b2 x 10b3 | 4b2 x 4b2 | 4b2 x 6b | 4b2 x 8 |
and then this
| 10b3 | 4b2 | 6b | 8 | |
| 50 | 500b3 | 200b2 | 300b | 300 |
| 6b | 60b4 | 24b3 | 36b2 | 48b |
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
The foregoing could have been done in one step. Now instead of add the intermediate rows by rows, we will add them along the diagonals in the powers of b are identical.
If we have made no mistakes, the foregoing gives the result P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3)
= 300 + (300 + 48)b + (200 + 36 + 32)b2 + (500+24+24)b3 + (60+16)b4 + 40b5
= 300 + 348b + 268b2 + 548b3 + 76b4 + 40b5
Whether or not the powers of b increase or decrease in the result is a cosmetic convention - some prefer one way, others the other, and some either.
Our conclusion follows. The product
P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3) =
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
That is computation of the left hand side for a given value of b gives the same result as computation of the right.
C . Multiplication Table Approach -
Developing a column multiplication method for polynomials
The above geometric approach suggests a table method:
| × | 10b3 | 4b2 | 6b | 8 |
| 50 | 500b3 | 200b2 | 300b | 300 |
| 6b | 60b4 | 24b3 | 36b2 | 48b |
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
which holds for real number b as well as whole numbers since the generalized distributive law holds for real numbers as well as whole numbers (why to come later). Here the rows and columns of the table come from the terms in the factors of the product to be computed, here (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3) while the entries inside list or tabulate the products of pairs of terms, one from each factor.
D. Column Method for Multiplication of Polynomials
Here we modify the table approach and dedicate a column to each power of b as follows
| × | 10b3 | 4b2 | 6b | 8 | ||
| 50 | 500b3 | 200b2 | 300b | 300 | ||
| 6b | 60b4 | 24b3 | 36b2 | 48b | ||
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
In this modified table approach, we compute all possible products as before, but align the products in each row according to their power of b. That makes addition and collecting like powers of b (with the aid of the distributive law) simpler. We add an extra row for the sum.
| × | 10b3 | 4b2 | 6b | 8 | ||
| 50 | 500b3 | 200b2 | 300b | 300 | ||
| 6b | 60b4 | 24b3 | 36b2 | 48b | ||
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 | ||
| P= | 40b5 | + 76b4 | + 548b3 | + 268b2 | +348b | +300 |
So the product P = 300 + 348b + 268b2 + 548b3 + 76b4 + 40b5 as before
In retrospect, the table (and column multiplication below) will be easier to do if we arrange the powers of b in ascending (that is, increasing) order along the top row.
| × | 8 | 6b | 4b2 | 10b3 | ||
| 50 | 300 | 300b | 200b2 | 500b3 | ||
| 6b | 48b | 36b2 | 24b3 | 60b4 | ||
| 4b2 | 32b2 | 24b3 | 16b4 | 40b5 | ||
| P= | 300 | + 348b | + 268b2 | +548b3 | +76b4 | +40b5 |
and align the left column at the left of the page (or page margin), so that we may write from left to right. Here again all possible products appear, but aligned in columns.
Finally, let us introduce or switch to the table method for multiplication, a change of notation.
Column Method for Products of Polynomials
8 + 6 b
+ 4b2 + 10b3
50 + 6b + 4b2
×
300 + 300b + 200b2 + 500b3
(product of top row with 50)
48b + 36b2 + 24b3 +
60b4
(product of first row with 6b)
32b2 + 24b3 + 16b4
+ 40b5 (product of first row with 4b2)
------------------------------------------------------------------------ +
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
------------------------------------------------------------------------
Note: The ascending order appears to work best as the position of the intermediate products shifts to the right away from the left margin where the computation begins. Do you see how each entry in each row of the column method corresponds to an entry in the rectangular approach and the two preceding tabular approaches to the computation of the product? All is a consequence of the distributive seen algebraically or geometrically visualized.
E. Column Method for Addition
In the above calculation, we see how to add polynomials
(300 + 300b + 200b2 + 500b3 ) + (48b
+ 36b2 + 24b3 + 60b4 )
+ (32b2 + 24b3 + 16b4
+ 40b5 )
via an addition methods in which like terms are aligned in columns as follows.
300 + 300b + 200b2 + 500b3
48b + 36b2 + 24b3 +
60b4
32b2 + 24b3 + 16b4
+ 40b5
------------------------------------------------------------------------ +
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
------------------------------------------------------------------------
F. Column Method for Subtraction
The difference of two polynomials
(18 + 48b + 36b2 +
24b3 + 60b4) - (10 + 8 b+ 32b2
+ 30b3 + 16b4
)
can also be computed by a column method
18 + 48b + 36b2 + 24b3
+ 60b4
10 + 8 b + 32b2 + 30b3
+ 16b4 (subtract)
------------------------------------------------------------------------
8 + 40b + 4 b2 -
6b3 + 44b4
------------------------------------------------------------------------
Here, as the column addition method, we align like powers of b.
A more informal alignment is method is to write
(18 + 48b + 36b2 + 24b3 + 60b4) - (10 + 8 b+ 32b2 + 30b3 + 16b4 )
= (18 + 48b + 36b2 + 24b3 + 60b4)
- (10 + 8 b+ 32b2 + 30b3 + 16b4 )
= 8 + 40b + 4 b2 - 6b3 + 44b4
Saying how to compute a difference defines it.
| Digression: Multiplication,
Addition and Subtraction with Decimals
The case where the variable b is replaced by the number 10 leads to a justification of the column method for multiplication of whole numbers using their decimal representation with powers of 10 written in decreasing order rather than increasing, Some conversion need to be considered. See Distributive Law Consequences |
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SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!