Problem 18 - Algebra Form and Function

Question

Is the expression a polynomial in the given variable? $$ P\left(1+\frac{r}{12}\right)^{10}, \text { in } r $$

Step-by-Step Solution

Answer

Yes, the expression P(1+(r/12))^10 is a polynomial in the variable r, as the expansion of this expression using the binomial theorem will result in terms with r raised to non-negative integer powers.

Step 1: Identify the given expression

The given expression is: $$ P\left(1+\frac{r}{12}\right)^{10}, \text { in } r $$ We will analyze this expression to determine if it is a polynomial in r.

Step 2: Check if the powers of r are non-negative integers

A polynomial consists of terms with non-negative integer powers. In this expression, only one term has the variable r, which is: $$ \frac{r}{12} $$ The power of r in this term is 1, which is a non-negative integer. Let's now examine the entire expression.

Step 3: Analyze the entire expression

In the expression, the term containing r is raised to the 10th power: $$ \left(1+\frac{r}{12}\right)^{10} $$ Recall that when we expand a binomial expression raised to a power (using the binomial theorem), it takes the form of a sum of the powers of binomial terms. So, when we expand the given expression, we will obtain a sum of terms in which the powers of r are non-negative integers. As a result, this expression, when expanded, will form a polynomial in r.

Step 4: Conclusion

Since the powers of r in the expanded expression will always be non-negative integers, the expression $$ P\left(1+\frac{r}{12}\right)^{10} $$ is a polynomial in the variable r.

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra. It gives us a powerful way to expand expressions of the form \((a + b)^n\). This theorem states that such an expression can be expanded as a sum of terms in the form \( \binom{n}{k} a^{n-k}b^k \), where \( k \) is a non-negative integer and \( \binom{n}{k} \) is a binomial coefficient.
The theorem is especially useful when working with large powers. Instead of manually multiplying the binomial, you can use the Binomial Theorem to write the expanded form quickly. This is known as binomial expansion.

• For example, if you have the expression \((x + y)^3\), you can expand it using the theorem:
• It becomes \(x^3 + 3x^2y + 3xy^2 + y^3\).

When it comes to the expression \( \left(1 + \frac{r}{12}\right)^{10} \), the Binomial Theorem will help in expanding it into a series of terms where each term will involve powers of \( r \). This means the expansion will provide a polynomial expression in the variable \( r \).

Non-negative Integer Powers

In the study of polynomials, one crucial aspect is that the powers of the variables in a polynomial must be non-negative integers. This means that every exponent in a polynomial is a whole number greater than or equal to zero.

• An exponent like 2 or 5 works perfectly well because they are non-negative integers.
• However, a fractional or negative exponent would disqualify an expression from being a polynomial.

In the context of the expression \( \left(1+\frac{r}{12}\right)^{10} \), once expanded using the Binomial Theorem, each resulting term will have \( r \) raised to a power that is a non-negative integer.
This ensures the expanded form is a polynomial in \( r \). For example, possible terms in the expansion like \( r^2 \) or \( r^5 \) have clearly non-negative integer powers.

Polynomial Expression

A polynomial expression is a mathematical expression composed of variables, coefficients, and the operations of addition, subtraction, and non-negative integer exponents on the variables.

• The general form of a polynomial in one variable \( x \) is \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).
• Here, \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients, and \( n \) is a non-negative integer.

When considering an expression like \( P \left(1+\frac{r}{12}\right)^{10} \), after applying the Binomial Theorem, it results in a sum of terms that fit this definition of a polynomial.
Each term, when expanded, involves \( r \) raised to a non-negative integer power. This aligns perfectly with what constitutes a polynomial expression. The expanded result is a sum of these terms, ensuring it is indeed a polynomial in \( r \).