Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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!
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.