Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To find the coefficient of the third term in the expansion of the binomial [tex]\((3x^2 + 2y^3)^4\)[/tex], we can use the binomial theorem. The binomial theorem states that:
[tex]\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \][/tex]
For [tex]\((3x^2 + 2y^3)^4\)[/tex], we identify:
[tex]\[ a = 3x^2, \quad b = 2y^3, \quad n = 4 \][/tex]
We need to find the third term of the expansion. In the binomial theorem, terms are indexed from [tex]\(k = 0\)[/tex] to [tex]\(k = n\)[/tex], so the third term corresponds to [tex]\(k = 2\)[/tex].
The general term in the expansion is given by:
[tex]\[ T(k) = \binom{4}{k} (3x^2)^{4-k} (2y^3)^k \][/tex]
Substituting [tex]\(k = 2\)[/tex]:
[tex]\[ T(2) = \binom{4}{2} (3x^2)^{4-2} (2y^3)^2 \][/tex]
Now, we calculate each part of this term:
1. Binomial Coefficient [tex]\(\binom{4}{2}\)[/tex]:
[tex]\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \][/tex]
2. Power of [tex]\(3x^2\)[/tex]:
[tex]\[ (3x^2)^{4-2} = (3x^2)^2 = 3^2 \cdot (x^2)^2 = 9x^4 \][/tex]
3. Power of [tex]\(2y^3\)[/tex]:
[tex]\[ (2y^3)^2 = 2^2 \cdot (y^3)^2 = 4y^6 \][/tex]
Combining these results:
[tex]\[ T(2) = 6 \times 9x^4 \times 4y^6 \][/tex]
[tex]\[ T(2) = 6 \times 36x^4 y^6 \][/tex]
[tex]\[ T(2) = 216x^4 y^6 \][/tex]
So, the third term in the expansion of [tex]\((3x^2 + 2y^3)^4\)[/tex] is:
[tex]\[ 216x^4 y^6 \][/tex]
Therefore, the coefficient of the third term in the expansion of the binomial [tex]\((3x^2 + 2y^3)^4\)[/tex] is [tex]\(216\)[/tex].
[tex]\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \][/tex]
For [tex]\((3x^2 + 2y^3)^4\)[/tex], we identify:
[tex]\[ a = 3x^2, \quad b = 2y^3, \quad n = 4 \][/tex]
We need to find the third term of the expansion. In the binomial theorem, terms are indexed from [tex]\(k = 0\)[/tex] to [tex]\(k = n\)[/tex], so the third term corresponds to [tex]\(k = 2\)[/tex].
The general term in the expansion is given by:
[tex]\[ T(k) = \binom{4}{k} (3x^2)^{4-k} (2y^3)^k \][/tex]
Substituting [tex]\(k = 2\)[/tex]:
[tex]\[ T(2) = \binom{4}{2} (3x^2)^{4-2} (2y^3)^2 \][/tex]
Now, we calculate each part of this term:
1. Binomial Coefficient [tex]\(\binom{4}{2}\)[/tex]:
[tex]\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \][/tex]
2. Power of [tex]\(3x^2\)[/tex]:
[tex]\[ (3x^2)^{4-2} = (3x^2)^2 = 3^2 \cdot (x^2)^2 = 9x^4 \][/tex]
3. Power of [tex]\(2y^3\)[/tex]:
[tex]\[ (2y^3)^2 = 2^2 \cdot (y^3)^2 = 4y^6 \][/tex]
Combining these results:
[tex]\[ T(2) = 6 \times 9x^4 \times 4y^6 \][/tex]
[tex]\[ T(2) = 6 \times 36x^4 y^6 \][/tex]
[tex]\[ T(2) = 216x^4 y^6 \][/tex]
So, the third term in the expansion of [tex]\((3x^2 + 2y^3)^4\)[/tex] is:
[tex]\[ 216x^4 y^6 \][/tex]
Therefore, the coefficient of the third term in the expansion of the binomial [tex]\((3x^2 + 2y^3)^4\)[/tex] is [tex]\(216\)[/tex].
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.