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Chapter 6: Problem 58

Writing Explain why the terms of \((a-4)^{6}\) have alternating positive and negative signs.

Short Answer

Expert verified
In the expansion of \begin{align*}(a-4)^6\end{align*},the terms have alternating signs because the exponent on \(-4\) alternates between even and odd, affecting the sign due to even powers resulting in a positive sign and odd powers resulting in a negative sign.

Step by step solution

01

Understand the Binomial Theorem

The binomial theorem states that \begin{align*}(x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k\end{align*}where \(\binom{n}{k}\) is the binomial coefficient. The sign of each term in the expansion is determined by the sign of \(x\) and \(y\) being raised to different powers.
02

Identify the components related to \begin{align*}(a-4)^6\end{align*}

In this case, \(x=a\) and \(y=-4\). Since \(y\) is negative, raising \(-4\) to an even power will result in a positive number, and raising \(-4\) to an odd power will result in a negative number. This alternation of even and odd powers will lead to the alternating positive and negative signs.
03

Expand the binomial using the components

When \begin{align*}(a-4)^6\end{align*}is expanded using the binomial theorem, the exponents on \(-4\) will alternate between even and odd from one term to the next. Even powers keep the sign of \(-4\) as positive, and odd powers change it to negative, leading to alternating signs in the terms of the expanded binomial.

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Key Concepts

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

Binomial Expansion
At the core of understanding algebraic expressions lies the concept of binomial expansion. This allows us to express the power of a binomial鈥攖hat is, a two-term algebraic expression鈥� as a sum of terms involving coefficients and powers of the two variables. For example, the binomial expansion of

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Most popular questions from this chapter

a. Expand \((1+i)^{4} .\) b. Verify that \(1-i\) is a fourth root of \(-4\) by repeating the process in part (a) for \((1-i)^{4} .\)

Simplify each expression. $$ \frac{7 !}{3 !(7-3) !} $$

Use the Binomial Theorem to expand each binomial. $$ (x-1)^{6} $$

Expand each binomial. $$ (2 x-2 y)^{6} $$

Simplify each expression. $$ _{5} \mathrm{C}_{2}+_{5} \mathrm{C}_{3} $$
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