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To solve this problem using the **Equality of Polynomials Theorem**, we need to equate the coefficients of like powers of \( x \) on both sides of the given equation. Let's go through the steps to set up the resulting system of equations.

The given equation is:
\[
16x + 34 = (2A + 2B)x + (4A + 2B)
\]

### Step 1: Expanding and Equating Coefficients

The equation can be split based on the coefficients of \( x \) and the constant terms:
1. **Coefficient of \( x \):** 
   - On the left side, the coefficient of \( x \) is \( 16 \).
   - On the right side, the coefficient of \( x \) is \( 2A + 2B \).

   So, we have:
   \[
   16 = 2A + 2B
   \]

2. **Constant term:**
   - On the left side, the constant term is \( 34 \).
   - On the right side, the constant term is \( 4A + 2B \).

   So, we have:
   \[
   34 = 4A + 2B
   \]

### Step 2: Write the Resulting System of Equations

The system of equations is:
\[
\begin{cases}
16 = 2A + 2B \\
34 = 4A + 2B
\end{cases}
\]

Now we have the system in a form where we can solve for \( A \) and \( B \).

Would you like to proceed with solving this system of equations, or is the setup sufficient for now?

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Here are some related questions to extend your understanding:

1. How would you solve this system of equations to find values for \( A \) and \( B \)?
2. Can the Equality of Polynomials Theorem be applied to equations with higher powers of \( x \)?
3. How does the structure of a polynomial help in forming systems of equations?
4. What is the significance of equating coefficients in polynomial equations?
5. How does this method apply to solving polynomial identities?

**Tip:** Always check for consistency in the system of equations before solving—if the equations are dependent, they might not provide unique solutions.

Using the Equality of Polynomials Theorem, equate coefficients of like powers in the equation 16x + 34 = (2A + 2B)x + (4A + 2B). Write the resulting system of equations.

Learn how to equate coefficients in the polynomial equation 16x + 34 = (2A + 2B)x + (4A + 2B) to find the values of A and B. This solution covers key algebraic concepts and uses the Equality of Polynomials Theorem to create a system of equations. Suitable for students in Grades 9-11.

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