/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Expand \(\left(3 x^{2}+y\right)^... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 24

Expand \(\left(3 x^{2}+y\right)^{5}\)

Short Answer

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The short answer is: \(\left(3 x^{2}+y\right)^{5} = 243 x^{10} + 405 x^8y + 270 x^6y^2 + 90 x^4y^3 + 15 x^2y^4 + y^5\)

Step by step solution

01

Identify the Variables and Exponent

In this case, we have: - \(a = 3x^2\) - \(b = y\) - \(n = 5\) Our goal is to expand \(\left(3 x^{2}+y\right)^{5}\) using the binomial theorem.
02

Apply Binomial Theorem Formula

According to the binomial theorem, the expression \((a+b)^n\) can be expanded as: \[(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}\] In our case, \((a+b)^n = (3x^2 + y)^5\), so we need to find the expanded form using the formula above.
03

Calculate Binomial Coefficients and Expand Terms

Now, we can expand each term in the sum by calculating the binomial coefficients and substituting \(a\) and \(b\): \begin{align*} \left(3 x^{2}+y\right)^{5} &= \sum_{k=0}^{5} {5 \choose k} (3x^2)^{5-k} y^{k} \\ &= {5 \choose 0} (3x^2)^{5} y^{0}+ {5 \choose 1} (3x^2)^{4} y^{1} + {5 \choose 2} (3x^2)^{3} y^{2} + {5 \choose 3} (3x^2)^{2} y^{3} + {5 \choose 4} (3x^2)^{1} y^{4} + {5 \choose 5} (3x^2)^{0} y^{5} \end{align*}
04

Calculate Each Term's Coefficient

Now, let's simplify each of the terms: \begin{align*} {5 \choose 0} (3x^2)^{5} y^{0} &= 1 \cdot 243 x^{10} \cdot 1 = 243 x^{10} \\ {5 \choose 1} (3x^2)^{4} y^{1} &= 5 \cdot 81 x^8 \cdot y = 405 x^8y \\ {5 \choose 2} (3x^2)^{3} y^{2} &= 10 \cdot 27 x^6 \cdot y^2 = 270 x^6y^2 \\ {5 \choose 3} (3x^2)^{2} y^{3} &= 10 \cdot 9 x^4 \cdot y^3 = 90 x^4y^3 \\ {5 \choose 4} (3x^2)^{1} y^{4} &= 5 \cdot 3 x^2 \cdot y^4 = 15 x^2y^4 \\ {5 \choose 5} (3x^2)^{0} y^{5} &= 1 \cdot 1 \cdot y^5 = y^5 \end{align*}
05

Combine the Terms

Finally, we can combine all the terms to obtain the expanded form of the expression: \[\left(3 x^{2}+y\right)^{5} = 243 x^{10} + 405 x^8y + 270 x^6y^2 + 90 x^4y^3 + 15 x^2y^4 + y^5\] The expanded form of the given expression is: \[\boxed{243 x^{10} + 405 x^{8}y + 270 x^{6}y^{2} + 90 x^{4}y^{3} + 15 x^{2}y^{4} + y^5}\]

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