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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.

User Guide

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figure-out, &
learn.

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User Guide

To make best use of nubtrek, understand what is available.

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

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

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.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Polynomial Arithmetics

Voice

Voice

Home

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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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

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'

Progress

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
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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