High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Question 8 (5 points)

Use Pascal's triangle to expand each binomial.

[tex]\[(x-5y)^5\][/tex]

A. [tex]\[y^5 - 5y^4 x + 25y^3 x^2 - 125y^2 x^3 + 625y x^4 - 3125 x^5\][/tex]

B. [tex]\[x^5 + 5x^4 y + 10x^3 y^2 + 10x^2 y^3 + 5x y^4 + y^5\][/tex]

C. [tex]\[x^5 - 25x^4 y + 250x^3 y^2 - 1250x^2 y^3 + 3125x y^4 - 3125 y^5\][/tex]

D. [tex]\[x^5 - 5x^4 y + 25x^3 y^2 - 125x^2 y^3 + 625x y^4 - 3125 y^5\][/tex]

Answer :

Sure! Let's expand the binomial [tex]\((x - 5y)^5\)[/tex] using Pascal's Triangle.

Here's a step-by-step process:

1. Identify Pascal's Triangle Coefficients:
For expanding [tex]\((a + b)^n\)[/tex], we use the [tex]\(n\)[/tex]th row of Pascal's Triangle. For [tex]\(n = 5\)[/tex], the coefficients are:
[tex]\[
\binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \binom{5}{5}
\][/tex]
These coefficients are [tex]\(1, 5, 10, 10, 5, 1\)[/tex].

2. Write the General Form of the Expansion:
The binomial expansion of [tex]\((x - 5y)^5\)[/tex] is:
[tex]\[
\sum_{k=0}^{5} \binom{5}{k} (x)^{5-k} (-5y)^k
\][/tex]

3. Calculate Each Term:

- For [tex]\(k = 0\)[/tex]:
[tex]\[
\binom{5}{0} x^{5-0} (-5y)^0 = 1 \cdot x^5 \cdot 1 = x^5
\][/tex]

- For [tex]\(k = 1\)[/tex]:
[tex]\[
\binom{5}{1} x^{5-1} (-5y)^1 = 5 \cdot x^4 \cdot (-5y) = -25x^4y
\][/tex]

- For [tex]\(k = 2\)[/tex]:
[tex]\[
\binom{5}{2} x^{5-2} (-5y)^2 = 10 \cdot x^3 \cdot (25y^2) = 250x^3 y^2
\][/tex]

- For [tex]\(k = 3\)[/tex]:
[tex]\[
\binom{5}{3} x^{5-3} (-5y)^3 = 10 \cdot x^2 \cdot (-125y^3) = -1250x^2 y^3
\][/tex]

- For [tex]\(k = 4\)[/tex]:
[tex]\[
\binom{5}{4} x^{5-4} (-5y)^4 = 5 \cdot x^1 \cdot 625y^4 = 3125x y^4
\][/tex]

- For [tex]\(k = 5\)[/tex]:
[tex]\[
\binom{5}{5} x^{5-5} (-5y)^5 = 1 \cdot 1 \cdot (-3125y^5) = -3125y^5
\][/tex]

4. Combine All Terms Together:
[tex]\[
x^5 - 25x^4y + 250x^3y^2 - 1250x^2y^3 + 3125xy^4 - 3125y^5
\][/tex]

So, the expanded form of [tex]\((x - 5y)^5\)[/tex] is:
[tex]\[
x^5 - 25x^4y + 250x^3y^2 - 1250x^2y^3 + 3125xy^4 - 3125y^5
\][/tex]

This matches option [tex]\( \mathbf{x^5 - 25 x^4 y + 250 x^3 y^2 - 1250 x^2 y^3 + 3125 x y^4 - 3125 y^5} \)[/tex].