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Thought-Process to Discover Knowledge

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

  nub,

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nub is the simple explanation of the concept.

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Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

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exercise provides practice problems to become fluent in the concepts.

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summary of this topic

Polynomial Arithmetics

Polynomial Arithmetics

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Multiplication of Polynomials


 »  Numerical expressions can be multiplied
the multiplication distributes over addition or subtraction

    →  `[2^2+3(2)+1]` `xx[2^3]` `=2^5+3(2^4)+2^3`

    →  `[3^2+3]` `xx[3^3-5]` `=3^5+3^3-5(3^2)-15`


 »  Polynomial multiplication follows the numerical expression multiplication laws
    →  `[x^2+3(x)+1]` `xx[x^3]` `=x^5+3(x^4)+x^3`

    →  `[y^2+3]` `xx[y^3-5]` `=y^5+y^3-5(y^2)-15`

Multiplication of Polynomials

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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In this page, multiplication of polynomials is introduced.


Keep tapping on the content to continue learning.
In this page, multiplication of polynomials is introduced.

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`

The answer is '`20`'.

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

  • 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 can be handled as if the sub-expression is a number.

which of the following equals `4xx(3+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`

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`
  • `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`

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`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • `x^2-3y-4`
  • `x^2y-4x^2-3y+12`

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

your progress details

Progress

About you

Progress

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 4 times 5?
45
45
20
20
The answer is ' 20 '.
Given the numbers 4 and 5 are integers, What is 4 times 5.
1
2
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 it is a number.
Which of the following equals 4 times, 3 + 2.
1
2
The answer is 4 times 3, plus, 4 times 2
Note the following. 4 times 3 plus 2 = 4 times 5 = 20 ;; and 4 times 3 plus 4 times 2 = 12 + 8 =20 ;; This can be generalized for any variables ;; x multiplied y + z, = x multiplied y, + x multiplied z. This is an important result to work with polynomials. The variables x, y, z in this can be sub-expression of a polynomial. ;; An example is given with an algebraic expression a cube plus b squared.
multiply x and x squared + 3.
1
2
The answer is x cube plus 3x. The multiplication is carried out as follows.
To understand multiplication of polynomial, one such multiplication is illustrated.
Which of the following equals 4 times, 3 minus 2?
1
2
The answer is 4 times 3, minus, 4 times 2
Note the following. 4 times 3 minus 2 = 4 times 1 = 4 ;; and 4 times 3 minus 4 times 2 = 12 minus 8 =4 ;; This can be generalized for any variables ;; x multiplied y minus z, = x multiplied y, minus x multiplied z.
Note the given results. By substituting z 1 equals minus z 2 in the first result, the second result is derived. So these two are equivalent results.
multiply x and x squared minus 3.
1
2
The answer is x cube minus 3x. The multiplication is carried out as follows.
To understand multiplication of polynomial, one such multiplication is illustrated.
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. An example and sub expressions of the given example are shown. ;; Distributive property of multiplication over addition, x multiplied y plus z = x y + x z. ;; In this property, x, y, z can be any of the polynomial or the sub-expression as given above.
Which of the following equals x squared minus 3, multiplied, y minus 4?
1
2
The answer is "x squared y minus 4 x squared minus 3 y + 12"

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