Introduction

Understanding how to find the sum of coefficients in polynomial expansions is a fundamental skill in algebra and polynomial theory. This article will guide you through the process using the polynomial x2yZ10. We will break down the methodology and provide detailed explanations to ensure a thorough understanding.

Step-by-Step Guide to Finding the Sum of Coefficients

The sum of coefficients in a polynomial expression can be found by substituting the variables with 1. This method simplifies the expression to a straightforward calculation. Let's apply this approach to the polynomial x2yZ10.

Substitution Method and Calculation

To find the sum of coefficients in the expansion of x2yZ10, we need to substitute x 1, y 1, and Z 1 into the expression. This simplifies the polynomial to:

[(1 cdot 2 cdot 1)^{10} 2^{10}]

Next, we calculate the value of (2^{10}): [2^{10} 1024]

However, we need to consider the coefficients carefully. The correct substitution should be as follows:

[(1 cdot 2 cdot 1)^{10} 4^{10}]

Now, let's calculate (4^{10}): [4^{10} 2^{20} 1048576]

Generalizing the Method

The method works for any polynomial of the form (a_1x_1a_2x_2a_3x_3 cdots a_nx_n^m). By substituting (x_1 x_2 cdots x_n 1), we can simplify the expression to find the sum of the coefficients. For example, in the polynomial (ab^2), substituting (a 1) and (b 1), we get: [(1 cdot 1^2) 1]

The given polynomial (x2yZ^{10}) can be similarly simplified by substituting 1 for each variable:

[(1 cdot 2 cdot 1)^{10} (2 cdot 1)^{10} 2^{10} 4^{10}]

Important Points to Remember

Always substitute each variable with 1 to find the sum of coefficients in a polynomial. The substitution method works for any polynomial, regardless of its complexity. Calculate the simplified expression carefully to ensure accuracy.

Conclusion

In conclusion, to find the sum of coefficients in the expansion of x2yZ10, we followed a systematic approach by substituting x 1, y 1, and Z 1, which simplified the expression to 4^{10}. The final sum of the coefficients is 1048576. Understanding this method will help in solving similar problems efficiently.

Frequently Asked Questions (FAQs)

Q: Why do we substitute 1 for each variable? A: Substituting 1 for each variable effectively removes the variables and retains only the coefficients, simplifying the expression to a basic arithmetic operation. Q: Can this method be applied to non-polynomial expressions? A: No, this method is specific to polynomials and cannot be directly applied to non-polynomial expressions. Q: What is the sum of the coefficients for the polynomial (ab^3)? A: By substituting (a 1) and (b 1), we get (1^3 1), so the sum of the coefficients is 1.