Problem 74 - College Algebra

Question

In Exercises 73-76, determine whether each statement makes sense or does not make sense, and explain your reasoning. Without writing the expansion of \((x-1)^{6}\). I can see that the terms have alternating positive and negative signs.

Step-by-Step Solution

Answer

The statement that 'Without writing the expansion of \( (x-1)^6 \), I can see that the terms have alternating positive and negative signs' does make sense, as due to the properties of the binomial theorem, the signs of the terms vary between positive and negative in alternating manner.

Step 1: Understanding Binomial Theorem

In general for a binomial \( (a+b)^n \), the expansion according to the binomial theorem is given as \( (a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{n-1}ab^{n-1} + b^n \) where \( \binom{n}{r} \) indicates the binomial coefficient 'n choose r'. This formula expands the binomial by taking each term one by one in each step.

Step 2: Applying Theorem to Specific Problem

In the specific case of \( (x-1)^6 \), a is 'x' and b is '-1'. Notice that b is a negative number. Considering the formula, as n changes, the power on b changes. Since b is negative, an even power on b would turn the term positive and an odd power on b would turn the term negative.

Step 3: Conclusion

Given this understanding, the claim that the terms have alternating positive and negative signs in the expansion of \( (x-1)^6 \) is indeed correct and it is due to the power on -1 (which changes in each term of the expansion according to the binomial theorem), sometimes being odd (resulting in a negative term) and sometimes being even (resulting in a positive term).

Key Concepts

Binomial Expansion

The binomial expansion is a way of expressing the power of a binomial, which is an algebraic expression with two terms. For instance, when we expand \( (x-1)^6 \) without multiplying everything out, we rely on the binomial theorem. This theorem provides a shortcut to detailed multiplication by using a pattern that emerges from the binomial coefficients.

Simply put, binomial expansion allows us to write out the expression in a series of terms which are easier to compute. These terms include the powers of each binomial component and the special numbers known as binomial coefficients. The alternating signs in the expansion of \( (x-1)^6 \) are due to the fact that every other coefficient multiplies by a negative one, a pattern predictable by the binomial theorem.

Binomial Coefficients

Binomial coefficients are the specific numbers that appear as the multipliers of the terms in the binomial expansion. They are represented by the symbol \(\binom{n}{r}\), which is read as 'n choose r'.

These coefficients tell us how many ways there are to choose a subset of 'r' elements from a larger set of 'n' elements without considering the order. In terms of binomial expansion, they determine the weight of each term in the expansion. For example, in the expansion of \( (x-1)^6 \), binomial coefficients will tell you the number of ways to pick terms from the expansion where \( x \) and -1 contribute to the final term. As the position changes, so does the binomial coefficient, affecting whether the sign is positive or negative based on the power of -1.

Algebraic Expressions

Algebraic expressions are terms that include numbers, operators, variables, and sometimes exponents like the one in \( (x-1)^6 \). They represent values we want to find or specify relationships between numbers and variables.

In the context of the binomial theorem, algebraic expressions form the base of what we are expanding. Here, the algebraic expression \( (x-1) \) acts as our binomial, with two terms, \( x \) and -1, which we want to raise to the sixth power. Algebraic expressions are essential since they are the foundation of forming polynomial expansions, effectively leading to various applications in mathematics as well as in real-world situations.

Polynomial Expansion

Polynomial expansion refers to the process of expanding an algebraic expression that involves multiple terms. When we apply the binomial theorem to expand a binomial, we essentially are creating a polynomial.

The expansion of \( (x-1)^6 \) yields a polynomial with several terms. Each term of the expansion conforms to the binomial theorem, involving both binomial coefficients and the respective powers of the terms in the original binomial. Polynomial expansion allows us to simplify expressions and calculations, and can also facilitate understanding patterns within algebraic structures. The binomial expansion is a specialized case of polynomial expansion where the original expression has exactly two terms.