Understanding the Sum of Coefficients in Polynomial Expansion

The question asks us to find the sum of the coefficients when the given expression is expanded. The expression is a product of two polynomials raised to certain powers: $$ (x - 2y + 3z)^4 \cdot (x^2 + y - z^3)^3 $$

A key shortcut for finding the sum of coefficients of any polynomial is to substitute the value 1 for each variable in the polynomial. If we have a polynomial $P(x_1, x_2, ..., x_n)$, the sum of its coefficients is simply $P(1, 1, ..., 1)$. Let's apply this rule to the given expression.

Applying the Rule to the Expression

Let the given expression be represented by $E(x, y, z)$. $$ E(x, y, z) = (x - 2y + 3z)^4 \cdot (x^2 + y - z^3)^3 $$

To find the sum of the coefficients, we set $x=1$, $y=1$, and $z=1$: $$ E(1, 1, 1) = (1 - 2(1) + 3(1))^4 \cdot (1^2 + 1 - 1^3)^3 $$

Step-by-Step Calculation of the Sum

We can calculate the value by evaluating each part of the expression separately.

Evaluating the First Factor: $ (x - 2y + 3z)^4 $

• Substitute $x=1$, $y=1$, $z=1$: $ (1 - 2(1) + 3(1)) $
• Simplify the expression inside the parenthesis: $ 1 - 2 + 3 = -1 + 3 = 2 $
• Raise the result to the power of 4: $ 2^4 $
• Calculate the final value: $ 2^4 = 16 $

Evaluating the Second Factor: $ (x^2 + y - z^3)^3 $

• Substitute $x=1$, $y=1$, $z=1$: $ (1^2 + 1 - 1^3) $
• Calculate the powers: $ 1^2 = 1 $ and $ 1^3 = 1 $
• Simplify the expression inside the parenthesis: $ 1 + 1 - 1 = 1 $
• Raise the result to the power of 3: $ 1^3 $
• Calculate the final value: $ 1^3 = 1 $

Calculating the Final Sum

Now, we multiply the results of the two factors: $$ \text{Sum of Coefficients} = (\text{Result of First Factor}) \times (\text{Result of Second Factor}) $$ $$ \text{Sum of Coefficients} = 16 \times 1 $$ $$ \text{Sum of Coefficients} = 16 $$

Conclusion

By substituting $x=1$, $y=1$, and $z=1$ into the expression $ (x - 2y + 3z)^4 \cdot (x^2 + y - z^3)^3 $, we found the sum of the coefficients to be 16. This corresponds to one of the options provided.