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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

In this page, multiplication of polynomials is introduced.



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Note: This page is for students at 8th grade level who are not introduced to closure, commutative, associative, and distributive laws of numerical arithmetic.

What is `4xx5`?

  • `45`
  • `20`
  • `20`

The answer is '`20`'.

Given the numbers `4` and `5` are integers, What is `4xx5`?

  • an integer
  • an integer
  • not an integer

The answer is 'an integer'.

If any two numbers are multiplied or added the result is another number.

In an algebraic expression, any sub-expression is a number. The sub-expression can be handled as if the it is a number.

which of the following equals `4xx(3+2)`?

  • `4 xx 3 + 4 xx 2`
  • `4 xx 3 + 4 xx 2`
  • `4xx3 + 2`

The answer is '`4 xx 3 + 4 xx 2`'

Note the following

 •  `4 xx color(deepskyblue)((3+2)) = 4 xx color(deepskyblue)5 = 20`

 •  `color(coral)(4 xx 3) + color(deepskyblue)(4 xx 2) = color(coral)(12) + color(deepskyblue)8 = 20`

This can be generalized for any variables

`color(coral)x xx color(deepskyblue)((y+z)) = color(coral)x xx color(deepskyblue)y + color(coral)x xx color(deepskyblue)z`.

This is an important result to work with polynomials. The variables x, y, z in this can be any sub-expression of a polynomial.

For example: if `x=a^3+b^2` then

`color(coral)x xx color(deepskyblue)((y+z)) = color(coral)x xx color(deepskyblue)y + color(coral)x xx color(deepskyblue)z`

is equivalently

`color(coral)((a^3+b^2)) xx color(deepskyblue)((y+z)) ``= color(coral)((a^3+b^2)) xx color(deepskyblue)y ``+ color(coral)((a^3+b^2)) xx color(deepskyblue)z`

Multiply `x` and `x^2+3`.

  • `x^3+3`
  • `x^3+3x`
  • `x^3+3x`

The answer is '`x^3+3x`'. The multiplication is carried out as follows.

Similar to `4 xx(3+2)` `=4xx3 + 4 xx2`, the multiplication of terms of polynomial are carried out,
`color(coral)(x) xx (color(deepskyblue)(x^2+3))` `= color(coral)(x) xx color(deepskyblue)(x^2) + color(coral)(x) xx color(deepskyblue)(3)`.

To understand multiplication of polynomial, one such multiplication is illustrated.

`(color(coral)(2x^2+3x+5))xx(color(deepskyblue)(x+7))`

sub-expression `(color(coral)(2x^2+3x+5))` is a number and holding it as a whole. Applying that number into `x+7`
`quad =(color(coral)(2x^2+3x+5))xx color(deepskyblue)(x)`
`quad quad quad + (color(coral)(2x^2+3x+5)) xx color(deepskyblue)(7)`

sub-expression `(color(coral)(2x^2+3x))` as a number
`quad =(color(coral)(2x^2+3x)) xx color(deepskyblue)(x) + color(coral)(5)xx color(deepskyblue)(x)`
`quad quad quad + (color(coral)(2x^2+3x)) xx color(deepskyblue)(7) +color(coral)(5) xx color(deepskyblue)(7)`

`quad =2x^3+3x^2+5x`
`quad quad quad + 14x^2+21x + 35`

`quad =2x^3+(3+14)x^2+(5+21)x+35`

`quad =2x^3+17x^2+26x+35`

which of the following equals `4xx(3-2)`?

  • `4 xx 3 - 4 xx 2`
  • `4 xx 3 - 4 xx 2`
  • `4xx3 - 2`

The answer is '`4 xx 3 - 4 xx 2`'

Note the following

 •  `4 xx color(deepskyblue)((3-2)) = 4 xx color(deepskyblue)1 = 4`

 •  `color(coral)(4 xx 3) - color(deepskyblue)(4 xx 2) = color(coral)(12) - color(deepskyblue)8 = 4`

This can be generalized for any variables

`color(coral)x xx color(deepskyblue)((y-z)) = color(coral)x xx color(deepskyblue)y - color(coral)x xx color(deepskyblue)z`.

Note the following

 •  `color(coral)(x_1) xx color(deepskyblue)((y_1+z_1)) ``= color(coral)(x_1) xx color(deepskyblue)(y_1) ``+ color(coral)(x_1) xx color(deepskyblue)(z_1)`

 •  `color(coral)(x_2) xx color(deepskyblue)((y_2-z_2)) ``= color(coral)(x_2) xx color(deepskyblue)(y_2) ``- color(coral)(x_2) xx color(deepskyblue)(z_2)`

By substituting `z_1 = -z_2` in the first result, the second result is derived. So these two are equivalent results. .

Multiply `x` and `x^2-3`.

  • `x^3-3`
  • `x^3-3x`
  • `x^3-3x`

The answer is '`x^3-3x`'. The multiplication is carried out as follows.

Similar to `4 xx(3-2)` `=4xx3 - 4 xx2`, the multiplication of terms of polynomial are carried out,
`color(coral)(x) xx (color(deepskyblue)(x^2-3))` `= color(coral)(x) xx color(deepskyblue)(x^2) - color(coral)(x) xx color(deepskyblue)(3)`.

To understand multiplication of polynomial, one such multiplication is illustrated.

`(color(coral)(2x^2-3x+5))xx(color(deepskyblue)(x-7))`

sub-expression `(color(coral)(2x^2-3x+5))` is a number and holding it as a whole. Applying that number into `x-7`
`quad =(color(coral)(2x^2-3x+5))xx color(deepskyblue)(x)`
`quad quad quad + (color(coral)(2x^2-3x+5)) xx color(deepskyblue)((-7))`

sub-expression `(color(coral)(2x^2-3x))` as a number
`quad =(color(coral)(2x^2-3x)) xx color(deepskyblue)(x) + color(coral)(5)xx color(deepskyblue)(x)`
`quad quad quad + (color(coral)(2x^2-3x)) xx color(deepskyblue)((-7)) ``+color(coral)(5) xx color(deepskyblue)((-7))`

`quad =2x^3-3x^2+5x`
`quad quad quad -14x^2+21x - 35`

`quad =2x^3+(-3-14)x^2`` +(5+21)x-35`

`quad =2x^3-17x^2+26x-35`

Polynomials as a whole or each of the sub expression can be considered as numbers.

Multiplication is performed by distribution of multiplication into addition or subtraction.

Multiplication of Polynomials: A polynomial is considered to be a number as a whole or the sub expressions as numbers.

Eg: `(a+b+c^3)xx(p^2-q)`
in this each of the following can be considered a number
`a`;
`b`;
`c^3`;
`a+b`;
`b+c^3`;
`a+c^3`;
`a+b+c^3`;
`p^2`;
`-q`;
`(p^2-q)`.

Distributive property of multiplication over addition
`x(y+z) = xy+xz`
In this property, `x`, `y`, `z` can be any of the polynomial or the sub-expression as given above.

Solved Exercise Problem:

Which of the following equals `(x^2-3)(y-4)`

  • `x^2-3y-4`
  • `x^2y-4x^2-3y+12`
  • `x^2y-4x^2-3y+12`

The answer is '`x^2y-4x^2-3y+12`'

                            
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