Key Concept: Sum of Coefficients in a Polynomial Expansion

The sum of all coefficients in the expansion of any polynomial function $P(x)$ can be easily found by substituting $x=1$ into the polynomial. This is because when $x=1$, any term $c_k x^k$ simply becomes $c_k (1)^k = c_k$. Thus, evaluating $P(1)$ yields the sum of all its coefficients.

Step-by-Step Derivation of A

1. Identify the Polynomial: We are given that A denotes the sum of all coefficients in the expansion of $\left(1-3 x+10 x^2\right)^n$. Let $P(x) = \left(1-3 x+10 x^2\right)^n$.

2. Apply the Key Concept: To find the sum of coefficients, which is A, we substitute $x=1$ into the polynomial $P(x)$. $A = \left(1-3(1)+10(1)^2\right)^n$

3. Simplify the Expression: Perform the arithmetic operations inside the parentheses. $A = (1 - 3 + 10)^n$ $A = (8)^n$ So, we have $A = 8^n$.

Step-by-Step Derivation of B

1. Identify the Polynomial: Similarly, B denotes the sum of all coefficients in the expansion of $\left(1+x^2\right)^n$. Let $Q(x) = \left(1+x^2\right)^n$.

2. Apply the Key Concept: To find the sum of coefficients, which is B, we substitute $x=1$ into the polynomial $Q(x)$. $B = \left(1+(1)^2\right)^n$

3. Simplify the Expression: Perform the arithmetic operations inside the parentheses. $B = (1+1)^n$ $B = (2)^n$ So, we have $B = 2^n$.

Establishing the Relationship between A and B

Now that we have expressions for A and B in terms of $n$: $A = 8^n$ $B = 2^n$ We need to find how A and B are related. Let's express $8^n$ using the base $2$. We know that $8 = 2^3$.

1. Substitute and Rewrite A: Replace $8$ with $2^3$ in the expression for A. $A = (2^3)^n$

2. Apply Exponent Rule: Using the exponent rule $(a^m)^n = a^{mn}$, we multiply the exponents. $A = 2^{3n}$

3. Relate to B: We can rewrite $2^{3n}$ as $(2^n)^3$. $A = (2^n)^3$ Since we found that $B = 2^n$, we can substitute B into this equation. $A = B^3$

Tips for Success and Common Pitfalls

• Universality: Remember that this method ($P(1)$) works for finding the sum of coefficients for any polynomial, not just binomial expansions.
• Careful Substitution: Double-check your arithmetic when substituting $x=1$, especially with negative signs.
• Distinguish from Binomial Theorem: While this problem involves binomial expansions, the core concept used (sum of coefficients) is distinct from finding specific terms or properties of binomial coefficients.
• Other Sums: If a question asks for the sum of coefficients of terms with even powers or odd powers, a different approach involving $P(1)$ and $P(-1)$ would be needed.

Summary and Key Takeaway

By applying the fundamental principle that the sum of coefficients of a polynomial $P(x)$ is simply $P(1)$, we efficiently calculated $A=8^n$ and $B=2^n$. Subsequently, recognizing that $8$ is $2^3$, we were able to establish the relationship $A = B^3$. This highlights the importance of understanding basic algebraic properties and exponent rules in simplifying and solving problems.