If p(x)=(3x+2)(2x-3), what are the zeros of the polynomial?

To find the zeros of the polynomial \( p(x) = (3x + 2)(2x - 3) \), we set \( p(x) = 0 \) and solve for \( x \).

### Step 1: Set the polynomial equal to zero
\[
(3x + 2)(2x - 3) = 0
\]

### Step 2: Solve for \( x \) using the zero-product property
The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we solve each factor separately:

1. \( 3x + 2 = 0 \)
   \[
   3x = -2
   \]
   \[
   x = \frac{-2}{3}
   \]

2. \( 2x - 3 = 0 \)
   \[
   2x = 3
   \]
   \[
   x = \frac{3}{2}
   \]

### Step 3: Conclusion
The zeros of the polynomial are:
\[
x = \frac{-2}{3} \quad \text{and} \quad x = \frac{3}{2}
\]

Would you like more details on this solution or have any questions?

Here are 5 related questions for practice:
1. What are the zeros of \( (4x - 5)(x + 7) \)?
2. How do you find the zeros of a quadratic function?
3. If \( f(x) = (5x + 1)(x - 4) \), what are the zeros of \( f(x) \)?
4. How do the factors of a polynomial relate to its graph?
5. What happens if both factors in a factored polynomial are the same?

**Tip:** Always check your solutions by plugging the zeros back into the original equation to verify.

Discover how to find the zeros of the polynomial p(x) = (3x + 2)(2x - 3) using the Zero-Product Property. This problem covers key algebraic concepts such as factoring and solving polynomial equations. Ideal for students in Grades 8-10.

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