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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.
mathsIntroduction to Algebra, Polynomials, and IdentitiesPolynomial Arithmetics

### Multiplication of Polynomials

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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

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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