Find the 5^("th ") term in the expansion of (5x-y)^(6) in simplest form. Answer:

Identify general term: Identify the general term of a binomial expansion. The general term in the expansion of (a - b)^n is given by T(k+1) = C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient ＂n choose k＂.Determine specific term: Determine the specific term we are looking for. We want to find the 5th term, which corresponds to k = 4, since the first term corresponds to k = 0.Apply binomial theorem: Apply the binomial theorem to find the 5th term. Using the formula from Step 1, we substitute n = 6 and k = 4 to find T(5) = C(6, 4) * (5x)^(6-4) * (-y)^4.Calculate binomial coefficient: Calculate the binomial coefficient C(6, 4). C(6, 4) = 6! / (4! * (6-4)!) = (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1)) = (6 * 5) / (2 * 1) = 15.Substitute values: Substitute the values into the term. T(5) = 15 * (5x)^2 * (-y)^4 = 15 * 25x^2 * y^4.Simplify term: Simplify the term. T(5) = 15 * 25 * x^2 * y^4 = 375 * x^2 * y^4.

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Find the
5^("th ") term in the expansion of
(5x-y)^(6) in simplest form.
Answer:

Find the $5^{\text {th }}$ term in the expansion of $(5 x-y)^{6}$ in simplest form. $\newline$ Answer:

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Math Problems

Algebra 1

Identify a sequence as explicit or recursive

Full solution

Q. Find the

5^{\text {th }}

term in the expansion of

(5 x-y)^{6}

in simplest form.

\newline

Answer:

Identify general term: Identify the general term of a binomial expansion. $\newline$ The general term in the expansion of $(a - b)^n$ is given by $T(k+1) = C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient ＂n choose k＂.
Determine specific term: Determine the specific term we are looking for. $\newline$ We want to find the $5^{\text{th}}$ term, which corresponds to $k = 4$ , since the first term corresponds to $k = 0$ .
Apply binomial theorem: Apply the binomial theorem to find the $5^{\text{th}}$ term. Using the formula from Step $1$ , we substitute $n = 6$ and $k = 4$ to find $T(5) = C(6, 4) \cdot (5x)^{6-4} \cdot (-y)^4$ .
Calculate binomial coefficient: Calculate the binomial coefficient $C(6, 4)$ . $C(6, 4) = \frac{6!}{4! \cdot (6-4)!} = \frac{(6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}{((4 \cdot 3 \cdot 2 \cdot 1) \cdot (2 \cdot 1))} = \frac{(6 \cdot 5)}{(2 \cdot 1)} = 15$ .
Substitute values: Substitute the values into the term. $\newline$ $T(5) = 15 \times (5x)^2 \times (-y)^4 = 15 \times 25x^2 \times y^4$ .
Simplify term: Simplify the term. $T(5) = 15 \times 25 \times x^2 \times y^4 = 375 \times x^2 \times y^4$ .