Problem 60 - Algebra 2

Question

Error Analysis A student claims that 7 \(\mathrm{C}_{5} p^{2} q^{4}\) is a term in a binomial expansion. Explain the student's error.

Step-by-Step Solution

Answer

The student's error is that the sum of the powers of p (2) and q (4) in the term \(7 \mathrm{C}_{5} p^{2} q^{4}\) does not equal 7, which is indicated by the binomial coefficient. In a correct term of a binomial expansion, the sum of the powers has to equal the power in the binomial expansion itself, which is not the case here.

Step 1: Understanding the term in the binomial expansion

The general term in the binomial expansion of \((p+q)^{n}\) is given by \( \mathrm{T}_{r+1} = ^nC_rp^{n−r}q^{r}\). Therefore, each term in a binomial expansion should always have the sum of the powers of p and q equal to n.

Step 2: Identify the problem with the student's term

The given term is \(7 \mathrm{C}_{5} p^{2} q^{4}\). Thus, the power of \(p\) is 2 and the power of \(q\) is 4. The sum of the powers is \(2+4=6\). However, the binomial coefficient points to the expansion of \((p+q)^{7}\). Therefore, there should be 7 powers in total. This is not the case in the student's term, so the term does not fit into the binomial expansion.

Key Concepts

Binomial Coefficient

In the realm of binomial expansions, one frequently encounters binomial coefficients. These special numbers are used to determine the coefficients of the terms in the expansion of a binomial expression raised to a power. When you expand \((p+q)^n\), each term in the expansion is multiplied by a specific binomial coefficient.

Binomial coefficients are denoted as \(^nC_r\) or \(\binom{n}{r}\), and they can be calculated using the formula: \[^nC_r = \frac{n!}{r!(n-r)!}\]where \(!\) indicates a factorial, which is the product of all positive integers up to that number. These coefficients are pivotal because they determine how many ways one can choose \(r\) elements from a set of \(n\) elements.

• They appear in Pascal's Triangle, a geometric configuration that conveniently displays the coefficients.
• They are symmetrical, meaning \(^nC_r\) is the same as \(^nC_{n-r}\).

Terms in Binomial Expansion

When expanding a binomial expression such as \((p+q)^n\), you want to know how the terms in the binomial expansion are formulated. Each term in the binomial expansion takes the form of a binomial coefficient multiplied by the variables raised to certain powers.

The expression for the \(r+1\)th term in the expansion of \((p+q)^n\) is: \[T_{r+1} = ^nC_r p^{n-r} q^{r}\]This succinct formula explains how each term is constructed:

• \(^nC_r\) is the binomial coefficient, indicating how many ways you can arrange the terms.
• \(p^{n-r}\) and \(q^r\) dictate the powers of \(p\) and \(q\) for the term.
• The sum of the powers of \(p\) and \(q\) should always equal \(n\), the power to which the binomial is raised.

This is crucial because it ensures that the terms properly correspond to the binomial's expansion, as was highlighted in the original student's error.

Combinations in Algebra

Combinations play a significant role in algebra, especially in the context of binomial expansions. In algebra, combinations are used to decide how items can be selected from a group, emphasizing the arrangement without regard to order.

Mathematically, the concept of combinations is used to calculate binomial coefficients, which are the heart of the binomial expansion.

• They help determine the coefficients of each term in the expanded form \((p+q)^n\), known as the binomial theorem.
• Combinatorial logic ensures the formulation of terms aligns with numeric patterns such as those found in Pascal's Triangle.

Understanding combinations enables students to better grasp why certain terms in a binomial expansion exist and how the order of operations affects algebraic expressions. This thorough understanding pinpoints why each \(^nC_r\) correctly matches the power distributions in expansions, and why missteps, like those in the student's error, occur when this logic isn't correctly followed.