Mr. Rothman is using the Binomial Theorem to find the binomial expansion of (a+b)^7 Which of the following is a term in the expansion?(1 point)
a. 21a^6b
b. a^4b^3
c. 21a^2b^5
d. 21a^2b^4
To find the term in the binomial expansion, we use the formula:
Term = (n choose k) * a^(n-k) * b^k
Here, n is the exponent of (a+b) (which is 7), k is the index of the term we want to find (starting from 0), a is the coefficient of a in the term, and b is the coefficient of b in the term.
Let's go through each option:
a. 21a^6b
Here, a^6 is the term raised to the 6th power, which is not present in the expansion. So, this is not a term in the expansion.
b. a^4b^3
Here, (n choose k) = (7 choose 3) = 35, a^(n-k) = a^(7-3) = a^4, and b^k = b^3. So, this is a valid term in the expansion.
c. 21a^2b^5
Here, a^2 is the term raised to the 2nd power, which is not present in the expansion. So, this is not a term in the expansion.
d. 21a^2b^4
Here, a^2 is the term raised to the 2nd power, which is not present in the expansion. So, this is not a term in the expansion.
Therefore, the answer is b. a^4b^3.
To find a term in the expansion of (a+b)^7 using the Binomial Theorem, we need to use the formula:
(a+b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,k)a^(n-k) b^k + ... + C(n,n)a^0 b^n,
where C(n,k) represents the binomial coefficient, given by C(n,k) = n! / (k!(n-k)!).
In this case, n = 7, so the expansion will have terms with k ranging from 0 to 7.
Now we can identify which option is a term in the expansion by comparing the exponents of a and b:
a. 21a^6b: This term has a exponent of 6 for a and 1 for b. (Not a term in the expansion)
b. a^4b^3: This term has a exponent of 4 for a and 3 for b. (Not a term in the expansion)
c. 21a^2b^5: This term has a exponent of 2 for a and 5 for b. (Not a term in the expansion)
d. 21a^2b^4: This term has a exponent of 2 for a and 4 for b. (Correct term in the expansion)
Therefore, the correct answer is option d. 21a^2b^4.
To find a term in the binomial expansion of (a+b)^7 using the Binomial Theorem, we need to use the formula:
(a+b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n)a^0 b^n
In this case, n is 7. We can determine the coefficients using the binomial coefficient formula C(n,r) = n! / (r!(n-r)!).
Now, let's calculate the coefficients and identify the term in the expansion:
Coefficient for the term with a^6b:
C(7,6) = 7! / (6!(7-6)!) = 7! / (6! 1!) = 7
So, the coefficient for the term with a^6b is 7.
Therefore, the correct term in the expansion is 7a^6b.
None of the options provided match 7a^6b, so none of the given options is a term in the expansion.