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Chapter 14: Problem 13

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(5 x-1)^{3}$$

Short Answer

Expert verified
The simplified form of the binomial \((5x - 1)^3\) is \(125x^3 - 75x^2 + 15x - 1\)

Step by step solution

01

Binomial Theorem

To expand the binomial, use the Binomial Theorem formula: \((a + b) ^ n = \sum_{k = 0}^{n} = \binom{n}{k} (a ^ {n - k}) (b ^ k)\). Here, \(a = 5x\), \(b = -1\) and \(n = 3\)
02

Calculate the Binomial Coefficients

Calculate the binomial coefficients for \(n = 3\). The binomial coefficients are \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\). The value of these coefficients are 1, 3, 3, and 1 respectively.
03

Subsitute into the Formula

Substitute \(a = 5x\), \(b = -1\), \(n = 3\) and the calculated binomial coefficients into the binomial theorem formula and simplify. \(\binom{3}{0}(5x)^3(-1)^0 + \binom{3}{1}(5x)^2(-1)^1 + \binom{3}{2}(5x)(-1)^2 + \binom{3}{3}(-1)^3\)
04

Simplify the Power of Terms

Simplify the power of each term: \(125x^3 - 75x^2 + 15x - 1\)

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Use the formula \(a_{n}=a_{1} 3^{n-1}\) to find the 7 th term of the sequence \(11,33,99,297, \dots\)

Expand and write the answer as a single logarithm with a coefficient of 1. $$\sum_{i=1}^{4} \log (2 i)$$

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x+2)^{6}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{2}(-1)^{i} 2^{i}=0$$
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