/*! 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

Expert verified
Using the binomial theorem, the expansion of \((3x^2 + y)^5\) is: \((3x^2 + y)^5 = 243x^{10} + 405x^8y + 270x^6y^2 + 90x^4y^3 + 15x^2y^4 + y^5\)

Step by step solution

01

Understand the binomial theorem

The binomial theorem states that for any positive integer \(n\) and any real numbers \(a\) and \(b\), \((a+b)^n = \sum_{k=0}^n {n\choose k}a^{n-k}b^k\), where \({n\choose k}\) is the binomial coefficient, which denotes the number of ways to choose \(k\) elements from a set of \(n\) elements, and can be calculated as \({n\choose k} = \frac{n!}{k! (n-k)!}\). Step 2: Identifying the values in the given expression
02

Identify the values in the given expression

For the given expression \((3x^2 + y)^5\), we have \(n=5\), \(a=3x^2\), and \(b=y\). Step 3: Calculate and write down each term of the expansion
03

Calculate each term of the expansion

According to the binomial theorem, we store each term in the sum. For \(k=0\): \({5\choose 0}(3x^2)^{5-0}y^0 = 1\cdot(243x^{10})\cdot 1 = 243x^{10}\) For \(k=1\): \({5\choose 1}(3x^2)^{5-1}y^1 = 5\cdot(81x^8)\cdot y = 405x^8y\) For \(k=2\): \({5\choose 2}(3x^2)^{5-2}y^2 = 10\cdot(27x^6)\cdot y^2 = 270x^6y^2\) For \(k=3\): \({5\choose 3}(3x^2)^{5-3}y^3 = 10\cdot(9x^4)\cdot y^3 = 90x^4y^3\) For \(k=4\): \({5\choose 4}(3x^2)^{5-4}y^4 = 5\cdot(3x^2)\cdot y^4 = 15x^2y^4\) For \(k=5\): \({5\choose 5}(3x^2)^{5-5}y^5 = 1\cdot 1\cdot y^5 = y^5\) Step 4: Write down the final expansion
04

Write down the final expansion

Summing up all the terms calculated in step 3, we have the following expansion: \((3x^2 + y)^5 = 243x^{10} + 405x^8y + 270x^6y^2+ 90x^4y^3 + 15x^2y^4 + y^5\)

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