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Chapter 14: Problem 10

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x+4)^{3}$$

Short Answer

Expert verified
The expansion of \((x+4)^{3}\) is \(x^{3} + 12x^{2} + 48x + 64\)

Step by step solution

01

Apply the Binomial Theorem

The general form of the Binomial Theorem for the cube of a binomial expression \( (a+b)^{3} \) is \( a^{3} + 3a^{2}b +3ab^{2} + b^{3} \). We'll use this form to expand \((x+4)^{3}\)
02

Substitute the values

In our expression, \( a = x \), and \( b = 4 \). So, substituting these into the formula, we get: \( a^{3} + 3a^{2}b +3ab^{2} + b^{3} = x^{3} + 3x^{2}*4 +3x*4^{2} + 4^{3} \).
03

Simplify the Result

Perform the multiplication in each term. Simplify the expression and get the final result. So, \( x^{3} + 3x^{2}*4 +3x*4^{2} + 4^{3} = x^{3} + 12x^{2} + 48x + 64 \).

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