Answer :
Absolutely! Let's work through each multiplication step-by-step.
### (i) [tex]\((3x + 5y)(6x - 2y)\)[/tex]
To multiply these two binomials, we distribute each term in the first binomial by each term in the second binomial:
[tex]\[ (3x + 5y)(6x - 2y) = 3x \cdot 6x + 3x \cdot (-2y) + 5y \cdot 6x + 5y \cdot (-2y) \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 3x \cdot 6x = 18x^2 \][/tex]
[tex]\[ 3x \cdot (-2y) = -6xy \][/tex]
[tex]\[ 5y \cdot 6x = 30xy \][/tex]
[tex]\[ 5y \cdot (-2y) = -10y^2 \][/tex]
Next, combine the like terms (specifically the [tex]\(xy\)[/tex] terms):
[tex]\[ 18x^2 - 6xy + 30xy - 10y^2 = 18x^2 + 24xy - 10y^2 \][/tex]
So, the result is:
[tex]\[ 18x^2 + 24xy - 10y^2 \][/tex]
### (ii) [tex]\(\left(5x^2y + 2y^3\right)\left(3y - 6x^2y\right)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (5x^2y + 2y^3)(3y - 6x^2y) = 5x^2y \cdot 3y + 5x^2y \cdot (-6x^2y) + 2y^3 \cdot 3y + 2y^3 \cdot (-6x^2y) \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 5x^2y \cdot 3y = 15x^2y^2 \][/tex]
[tex]\[ 5x^2y \cdot (-6x^2y) = -30x^4y^2 \][/tex]
[tex]\[ 2y^3 \cdot 3y = 6y^4 \][/tex]
[tex]\[ 2y^3 \cdot (-6x^2y) = -12x^2y^4 \][/tex]
Combine the like terms:
[tex]\[ 15x^2y^2 - 30x^4y^2 + 6y^4 - 12x^2y^4 = -30x^4y^2 + 15x^2y^2 - 12x^2y^4 + 6y^4 \][/tex]
So, the result is:
[tex]\[ -30x^4y^2 + 15x^2y^2 - 12x^2y^4 + 6y^4 \][/tex]
### (iii) [tex]\((6m - 5n)(2m^2 + 3mn + 6n^2)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (6m - 5n)(2m^2 + 3mn + 6n^2) = 6m \cdot 2m^2 + 6m \cdot 3mn + 6m \cdot 6n^2 + (-5n) \cdot 2m^2 + (-5n) \cdot 3mn + (-5n) \cdot 6n^2 \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 6m \cdot 2m^2 = 12m^3 \][/tex]
[tex]\[ 6m \cdot 3mn = 18m^2n \][/tex]
[tex]\[ 6m \cdot 6n^2 = 36mn^2 \][/tex]
[tex]\[ -5n \cdot 2m^2 = -10m^2n \][/tex]
[tex]\[ -5n \cdot 3mn = -15mn^2 \][/tex]
[tex]\[ -5n \cdot 6n^2 = -30n^3 \][/tex]
Combine the like terms:
[tex]\[ 12m^3 + 18m^2n - 10m^2n + 36mn^2 - 15mn^2 - 30n^3 = 12m^3 + 8m^2n + 21mn^2 - 30n^3 \][/tex]
So, the result is:
[tex]\[ 12m^3 + 8m^2n + 21mn^2 - 30n^3 \][/tex]
### (iv) [tex]\(\left(3x^2y + 5xy^2\right)\left(2x^2 - 6xy + 4y^2\right)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (3x^2y + 5xy^2)(2x^2 - 6xy + 4y^2) = 3x^2y \cdot 2x^2 + 3x^2y \cdot (-6xy) + 3x^2y \cdot 4y^2 + 5xy^2 \cdot 2x^2 + 5xy^2 \cdot (-6xy) + 5xy^2 \cdot 4y^2 \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 3x^2y \cdot 2x^2 = 6x^4y \][/tex]
[tex]\[ 3x^2y \cdot (-6xy) = -18x^3y^2 \][/tex]
[tex]\[ 3x^2y \cdot 4y^2 = 12x^2y^3 \][/tex]
[tex]\[ 5xy^2 \cdot 2x^2 = 10x^3y^2 \][/tex]
[tex]\[ 5xy^2 \cdot (-6xy) = -30x^2y^3 \][/tex]
[tex]\[ 5xy^2 \cdot 4y^2 = 20xy^4 \][/tex]
Combine the like terms:
[tex]\[ 6x^4y - 18x^3y^2 + 10x^3y^2 + 12x^2y^3 - 30x^2y^3 + 20xy^4 = 6x^4y - 8x^3y^2 - 18x^2y^3 + 20xy^4 \][/tex]
So, the result is:
[tex]\[ 6x^4y - 8x^3y^2 - 18x^2y^3 + 20xy^4 \][/tex]
These are the complete, expanded forms for all the given polynomials!
### (i) [tex]\((3x + 5y)(6x - 2y)\)[/tex]
To multiply these two binomials, we distribute each term in the first binomial by each term in the second binomial:
[tex]\[ (3x + 5y)(6x - 2y) = 3x \cdot 6x + 3x \cdot (-2y) + 5y \cdot 6x + 5y \cdot (-2y) \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 3x \cdot 6x = 18x^2 \][/tex]
[tex]\[ 3x \cdot (-2y) = -6xy \][/tex]
[tex]\[ 5y \cdot 6x = 30xy \][/tex]
[tex]\[ 5y \cdot (-2y) = -10y^2 \][/tex]
Next, combine the like terms (specifically the [tex]\(xy\)[/tex] terms):
[tex]\[ 18x^2 - 6xy + 30xy - 10y^2 = 18x^2 + 24xy - 10y^2 \][/tex]
So, the result is:
[tex]\[ 18x^2 + 24xy - 10y^2 \][/tex]
### (ii) [tex]\(\left(5x^2y + 2y^3\right)\left(3y - 6x^2y\right)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (5x^2y + 2y^3)(3y - 6x^2y) = 5x^2y \cdot 3y + 5x^2y \cdot (-6x^2y) + 2y^3 \cdot 3y + 2y^3 \cdot (-6x^2y) \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 5x^2y \cdot 3y = 15x^2y^2 \][/tex]
[tex]\[ 5x^2y \cdot (-6x^2y) = -30x^4y^2 \][/tex]
[tex]\[ 2y^3 \cdot 3y = 6y^4 \][/tex]
[tex]\[ 2y^3 \cdot (-6x^2y) = -12x^2y^4 \][/tex]
Combine the like terms:
[tex]\[ 15x^2y^2 - 30x^4y^2 + 6y^4 - 12x^2y^4 = -30x^4y^2 + 15x^2y^2 - 12x^2y^4 + 6y^4 \][/tex]
So, the result is:
[tex]\[ -30x^4y^2 + 15x^2y^2 - 12x^2y^4 + 6y^4 \][/tex]
### (iii) [tex]\((6m - 5n)(2m^2 + 3mn + 6n^2)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (6m - 5n)(2m^2 + 3mn + 6n^2) = 6m \cdot 2m^2 + 6m \cdot 3mn + 6m \cdot 6n^2 + (-5n) \cdot 2m^2 + (-5n) \cdot 3mn + (-5n) \cdot 6n^2 \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 6m \cdot 2m^2 = 12m^3 \][/tex]
[tex]\[ 6m \cdot 3mn = 18m^2n \][/tex]
[tex]\[ 6m \cdot 6n^2 = 36mn^2 \][/tex]
[tex]\[ -5n \cdot 2m^2 = -10m^2n \][/tex]
[tex]\[ -5n \cdot 3mn = -15mn^2 \][/tex]
[tex]\[ -5n \cdot 6n^2 = -30n^3 \][/tex]
Combine the like terms:
[tex]\[ 12m^3 + 18m^2n - 10m^2n + 36mn^2 - 15mn^2 - 30n^3 = 12m^3 + 8m^2n + 21mn^2 - 30n^3 \][/tex]
So, the result is:
[tex]\[ 12m^3 + 8m^2n + 21mn^2 - 30n^3 \][/tex]
### (iv) [tex]\(\left(3x^2y + 5xy^2\right)\left(2x^2 - 6xy + 4y^2\right)\)[/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (3x^2y + 5xy^2)(2x^2 - 6xy + 4y^2) = 3x^2y \cdot 2x^2 + 3x^2y \cdot (-6xy) + 3x^2y \cdot 4y^2 + 5xy^2 \cdot 2x^2 + 5xy^2 \cdot (-6xy) + 5xy^2 \cdot 4y^2 \][/tex]
Now, perform the multiplication for each pair:
[tex]\[ 3x^2y \cdot 2x^2 = 6x^4y \][/tex]
[tex]\[ 3x^2y \cdot (-6xy) = -18x^3y^2 \][/tex]
[tex]\[ 3x^2y \cdot 4y^2 = 12x^2y^3 \][/tex]
[tex]\[ 5xy^2 \cdot 2x^2 = 10x^3y^2 \][/tex]
[tex]\[ 5xy^2 \cdot (-6xy) = -30x^2y^3 \][/tex]
[tex]\[ 5xy^2 \cdot 4y^2 = 20xy^4 \][/tex]
Combine the like terms:
[tex]\[ 6x^4y - 18x^3y^2 + 10x^3y^2 + 12x^2y^3 - 30x^2y^3 + 20xy^4 = 6x^4y - 8x^3y^2 - 18x^2y^3 + 20xy^4 \][/tex]
So, the result is:
[tex]\[ 6x^4y - 8x^3y^2 - 18x^2y^3 + 20xy^4 \][/tex]
These are the complete, expanded forms for all the given polynomials!
