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Sure, let's solve each of these products step-by-step.
1. [tex]\((12x + 15)(12x - 15)\)[/tex]
This is a difference of squares form:
[tex]\[ (a + b)(a - b) = a^2 - b^2 \][/tex]
Here, [tex]\(a = 12x\)[/tex] and [tex]\(b = 15\)[/tex]. So,
[tex]\[ (12x + 15)(12x - 15) = (12x)^2 - 15^2 = 144x^2 - 225 \][/tex]
2. [tex]\(\left(7m^2 - 13\right)\left(7m^2 + 13\right)\)[/tex]
Again, this is a difference of squares form. Here, [tex]\(a = 7m^2\)[/tex] and [tex]\(b = 13\)[/tex]. So,
[tex]\[ \left(7m^2 - 13\right)\left(7m^2 + 13\right) = (7m^2)^2 - 13^2 = 49m^4 - 169 \][/tex]
3. [tex]\((5x + 4)^3\)[/tex]
This is a binomial expansion:
[tex]\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \][/tex]
Here, [tex]\(a = 5x\)[/tex] and [tex]\(b = 4\)[/tex]. So,
[tex]\[ (5x + 4)^3 = (5x)^3 + 3(5x)^2(4) + 3(5x)(4)^2 + (4)^3 = 125x^3 + 300x^2 + 240x + 64 \][/tex]
4. [tex]\((10x + 3)^2\)[/tex]
This is a binomial square form:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
Here, [tex]\(a = 10x\)[/tex] and [tex]\(b = 3\)[/tex]. So,
[tex]\[ (10x + 3)^2 = (10x)^2 + 2(10x)(3) + (3)^2 = 100x^2 + 60x + 9 \][/tex]
5. [tex]\((9x + 4y)^2\)[/tex]
Again, this is a binomial square form. Here, [tex]\(a = 9x\)[/tex] and [tex]\(b = 4y\)[/tex]. So,
[tex]\[ (9x + 4y)^2 = (9x)^2 + 2(9x)(4y) + (4y)^2 = 81x^2 + 72xy + 16y^2 \][/tex]
6. [tex]\(\left(13x^2 + 11\right)\left(13x^2 - 11\right)\)[/tex]
This is a difference of squares form. Here, [tex]\(a = 13x^2\)[/tex] and [tex]\(b = 11\)[/tex]. So,
[tex]\[ \left(13x^2 + 11\right)\left(13x^2 - 11\right) = (13x^2)^2 - 11^2 = 169x^4 - 121 \][/tex]
7. [tex]\((11y - 21)(11y + 21)\)[/tex]
Again, this is a difference of squares form. Here, [tex]\(a = 11y\)[/tex] and [tex]\(b = 21\)[/tex]. So,
[tex]\[ (11y - 21)(11y + 21) = (11y)^2 - 21^2 = 121y^2 - 441 \][/tex]
8. [tex]\((7w + 9k + 3v)^2\)[/tex]
This is a trinomial square expansion:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \][/tex]
Here, [tex]\(a = 7w\)[/tex], [tex]\(b = 9k\)[/tex], and [tex]\(c = 3v\)[/tex]. So,
[tex]\[ (7w + 9k + 3v)^2 = (7w)^2 + (9k)^2 + (3v)^2 + 2(7w)(9k) + 2(7w)(3v) + 2(9k)(3v) \][/tex]
Calculating each term:
[tex]\[ (7w)^2 = 49w^2, \quad (9k)^2 = 81k^2, \quad (3v)^2 = 9v^2 \][/tex]
[tex]\[ 2(7w)(9k) = 126wk, \quad 2(7w)(3v) = 42wv, \quad 2(9k)(3v) = 54kv \][/tex]
Putting it all together:
[tex]\[ (7w + 9k + 3v)^2 = 49w^2 + 81k^2 + 9v^2 + 126wk + 42wv + 54kv \][/tex]
So, the simplified form is:
[tex]\[ = 81k^2 + 54kv + 126kw + 9v^2 + 42vw + 49w^2 \][/tex]
The results are:
1. [tex]\(144x^2 - 225\)[/tex]
2. [tex]\(49m^4 - 169\)[/tex]
3. [tex]\(125x^3 + 300x^2 + 240x + 64\)[/tex]
4. [tex]\(100x^2 + 60x + 9\)[/tex]
5. [tex]\(81x^2 + 72xy + 16y^2\)[/tex]
6. [tex]\(169x^4 - 121\)[/tex]
7. [tex]\(121y^2 - 441\)[/tex]
8. [tex]\(81k^2 + 54kv + 126kw + 9v^2 + 42vw + 49w^2\)[/tex]
1. [tex]\((12x + 15)(12x - 15)\)[/tex]
This is a difference of squares form:
[tex]\[ (a + b)(a - b) = a^2 - b^2 \][/tex]
Here, [tex]\(a = 12x\)[/tex] and [tex]\(b = 15\)[/tex]. So,
[tex]\[ (12x + 15)(12x - 15) = (12x)^2 - 15^2 = 144x^2 - 225 \][/tex]
2. [tex]\(\left(7m^2 - 13\right)\left(7m^2 + 13\right)\)[/tex]
Again, this is a difference of squares form. Here, [tex]\(a = 7m^2\)[/tex] and [tex]\(b = 13\)[/tex]. So,
[tex]\[ \left(7m^2 - 13\right)\left(7m^2 + 13\right) = (7m^2)^2 - 13^2 = 49m^4 - 169 \][/tex]
3. [tex]\((5x + 4)^3\)[/tex]
This is a binomial expansion:
[tex]\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \][/tex]
Here, [tex]\(a = 5x\)[/tex] and [tex]\(b = 4\)[/tex]. So,
[tex]\[ (5x + 4)^3 = (5x)^3 + 3(5x)^2(4) + 3(5x)(4)^2 + (4)^3 = 125x^3 + 300x^2 + 240x + 64 \][/tex]
4. [tex]\((10x + 3)^2\)[/tex]
This is a binomial square form:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
Here, [tex]\(a = 10x\)[/tex] and [tex]\(b = 3\)[/tex]. So,
[tex]\[ (10x + 3)^2 = (10x)^2 + 2(10x)(3) + (3)^2 = 100x^2 + 60x + 9 \][/tex]
5. [tex]\((9x + 4y)^2\)[/tex]
Again, this is a binomial square form. Here, [tex]\(a = 9x\)[/tex] and [tex]\(b = 4y\)[/tex]. So,
[tex]\[ (9x + 4y)^2 = (9x)^2 + 2(9x)(4y) + (4y)^2 = 81x^2 + 72xy + 16y^2 \][/tex]
6. [tex]\(\left(13x^2 + 11\right)\left(13x^2 - 11\right)\)[/tex]
This is a difference of squares form. Here, [tex]\(a = 13x^2\)[/tex] and [tex]\(b = 11\)[/tex]. So,
[tex]\[ \left(13x^2 + 11\right)\left(13x^2 - 11\right) = (13x^2)^2 - 11^2 = 169x^4 - 121 \][/tex]
7. [tex]\((11y - 21)(11y + 21)\)[/tex]
Again, this is a difference of squares form. Here, [tex]\(a = 11y\)[/tex] and [tex]\(b = 21\)[/tex]. So,
[tex]\[ (11y - 21)(11y + 21) = (11y)^2 - 21^2 = 121y^2 - 441 \][/tex]
8. [tex]\((7w + 9k + 3v)^2\)[/tex]
This is a trinomial square expansion:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \][/tex]
Here, [tex]\(a = 7w\)[/tex], [tex]\(b = 9k\)[/tex], and [tex]\(c = 3v\)[/tex]. So,
[tex]\[ (7w + 9k + 3v)^2 = (7w)^2 + (9k)^2 + (3v)^2 + 2(7w)(9k) + 2(7w)(3v) + 2(9k)(3v) \][/tex]
Calculating each term:
[tex]\[ (7w)^2 = 49w^2, \quad (9k)^2 = 81k^2, \quad (3v)^2 = 9v^2 \][/tex]
[tex]\[ 2(7w)(9k) = 126wk, \quad 2(7w)(3v) = 42wv, \quad 2(9k)(3v) = 54kv \][/tex]
Putting it all together:
[tex]\[ (7w + 9k + 3v)^2 = 49w^2 + 81k^2 + 9v^2 + 126wk + 42wv + 54kv \][/tex]
So, the simplified form is:
[tex]\[ = 81k^2 + 54kv + 126kw + 9v^2 + 42vw + 49w^2 \][/tex]
The results are:
1. [tex]\(144x^2 - 225\)[/tex]
2. [tex]\(49m^4 - 169\)[/tex]
3. [tex]\(125x^3 + 300x^2 + 240x + 64\)[/tex]
4. [tex]\(100x^2 + 60x + 9\)[/tex]
5. [tex]\(81x^2 + 72xy + 16y^2\)[/tex]
6. [tex]\(169x^4 - 121\)[/tex]
7. [tex]\(121y^2 - 441\)[/tex]
8. [tex]\(81k^2 + 54kv + 126kw + 9v^2 + 42vw + 49w^2\)[/tex]
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