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Chapter 19: Problem 3

Expand and simplify the given expressions by use of the binomial formula. $$(t+4)^{3}$$

Short Answer

Expert verified
The expanded and simplified form is \\(t^3 + 12t^2 + 48t + 64\\).

Step by step solution

01

Identify the Binomial Formula

To expand \(t + 4\)^3\, we will use the binomial formula, which is given by \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) where \(a = t\), \(b = 4\), and \(n = 3\).
02

Calculate the Binomial Coefficients

Determine the binomial coefficients for \(n = 3\). These are computed as follows: \(\binom{3}{0} = 1, \), \(\binom{3}{1} = 3, \), \(\binom{3}{2} = 3, \), and \(\binom{3}{3} = 1\).
03

Expand Using the Formula

Substitute the values for \(a\), \(b\), and the calculated coefficients into the binomial formula: \( (t+4)^3 = \binom{3}{0} t^3 \cdot 4^0 + \binom{3}{1} t^2 \cdot 4^1 + \binom{3}{2} t^1 \cdot 4^2 + \binom{3}{3} t^0 \cdot 4^3\).
04

Simplify Each Term

Simplify each term separately: \(\binom{3}{0} t^3 \cdot 4^0 = t^3, \) \(\binom{3}{1} t^2 \cdot 4^1 = 12t^2, \) \(\binom{3}{2} t^1 \cdot 4^2 = 48t, \) \(\binom{3}{3} t^0 \cdot 4^3 = 64\).
05

Combine the Terms

Finally, add all the terms together to get the expanded expression: \(t^3 + 12t^2 + 48t + 64\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expansion
Polynomial expansion is a way to express a binomial raised to a power as a sum of terms. It involves breaking down expressions like \((t+4)^3\) into a series of simpler parts. This is essential in various fields of math where transforming expressions into polynomial form can simplify calculations or proofs.
  • When expanding polynomials, each term represents a part of the whole expression. In this case, \((t+4)^3\) becomes a combination of terms involving powers of \(t\) and constants derived from 4.
  • The binomial theorem provides a formulaic approach to perform this expansion: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
  • By substituting the values \(a = t\), \(b = 4\), and \(n = 3\), we get the expanded form.
Expanding a polynomial not only changes its looks but also sometimes reveals properties that were not obvious in the unexpanded form. This is why mastering this technique is fundamental for advancing in algebra.
Binomial Coefficients
Binomial coefficients are integral to the binomial theorem, as they determine the weight of each expanded term. They are the coefficients that appear in the binomial expansion and are written as \(\binom{n}{k}\).
  • Each coefficient is a number that represents the number of ways to choose \(k\) elements from \(n\).
  • The calculation of these coefficients is straightforward using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
  • For \((t+4)^3\), the coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).
These values are always symmetrical, as seen here with \(\binom{3}{1}\) and \(\binom{3}{2}\) being equal, which is a useful property when checking expansions. Understanding and computing these coefficients are crucial steps in simplifying binomial expansions.
Simplification of Expressions
Simplification involves reducing the expanded polynomial into a form that conveys all the necessary mathematical information with the fewest terms. It鈥檚 the final touch in the process of expansion, where redundant or complicated expressions get boiled down to basics.
  • Once the polynomial expansion is complete, each term needs to be simplified individually. For instance, the expression \(\binom{3}{1} t^2 \cdot 4^1\) simplifies to \(12t^2\).
  • It's important to accurately calculate powers and products of each term.
  • After simplifying each term individually (like \(t^3\) from \(\binom{3}{0} t^3 \cdot 4^0\) and \(64\) from \(\binom{3}{3} t^0 \cdot 4^3\)), combine them to form the final expression: \(t^3 + 12t^2 + 48t + 64\).
In algebra, the skill of simplifying expressions allows us to analyze and interpret results more efficiently. Simplified expressions are cleaner to work with and are often necessary for further calculations or applications in more complex mathematical problems.

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Most popular questions from this chapter

Find any of the values of \(a_{1}, d, a_{n}, n,\) or \(S_{n}\) that are missing for an arithmetic sequence. $$d=-3, n=3, a_{3}=-5.9$$

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Find the sum of the first \(n\) terms of the indicated geometric sequence with the given values. $$384,192,96, \ldots, n=7$$

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