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Chapter 14: Problem 12

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x+3 y)^{3}$$

Short Answer

Expert verified
The correct simplified form of the binomial \( (x + 3y)^3 \) is \( x^3 + 9x^2y + 27xy^2 + 27y^3 \).

Step by step solution

01

Identify the coefficients

From the given, we have \( a = x \), \( b = 3y \), and \( n = 3 \). These elements correspond to the elements in the formula of the binomial theorem, where \( a + b \) is raised to an exponent \( n \).
02

Apply the Binomial Theorem

By substituting \( a=x \), \( b=3y \), and \( n=3 \) into the formula stated in the analysis, we get: \[ (x + 3y)^3 = C(3,0)x^3*(3y)^0 + C(3,1)x^2*(3y)^1 + C(3,2)x^1*(3y)^2 + C(3,3)x^0*(3y)^3 \]
03

Computation

Calculating the combination numbers and simplifying the terms, the expression becomes: \[ = 1*x^3 + 3*x^2*3y + 3*x*(9y^2) + 1*(27y^3) = x^3 + 9x^2y + 27x*y^2 + 27*y^3 \]

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$$\text { Solve: } \log \left(x^{2}-25\right)-\log (x+5)=3$$

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