Factors of the following expression will be :(x²−6xy + 9y²) – 16

Q: Factors of the following expression will be :(x²−6xy + 9y²) – 16

To determine the factors of the expression $(x^2 - 6xy + 9y^2) - 16$, we need to analyze and simplify the expression step by step. 1. **Simplifying the Expression:** The expression $(x^2 - 6xy + 9y^2)$ is a perfect square trinomial. It can be rewritten as $(x - 3y)^2$ because: \[ (x - 3y)^2 = x^2 - 2 \cdot x \cdot 3y + (3y)^2 = x^2 - 6xy + 9y^2 \] Therefore, the original expression becomes: \[ (x - 3y)^2 - 16 \] 2. **Recognizing the Difference of Squares:** The expression $(x - 3y)^2 - 16$ is a difference of squares. It can be factored using the identity $a^2 - b^2 = (a - b)(a + b)$, where $a = (x - 3y)$ and $b = 4$. 3. **Applying the Difference of Squares Formula:** Using the formula, we have: \[ (x - 3y)^2 - 4^2 = ((x - 3y) - 4)((x - 3y) + 4) \] Simplifying further, we get: \[ (x - 3y - 4)(x - 3y + 4) \] 4. **Conclusion:** The factors of the expression $(x^2 - 6xy + 9y^2) - 16$ are $(x - 3y - 4)(x - 3y + 4)$, which corresponds to Option 3. **Explanation of Incorrect Options:** - **Option 1: $(x - 4)(x + 4)$** - This option suggests a difference of squares involving $x^2 - 16$, which does not match the original expression $(x^2 - 6xy + 9y^2) - 16$. - **Option 2: $(x - y - 4)(x - y + 4)$** - This option implies a difference of squares involving $(x - y)^2 - 16$, which does not align with the expression $(x - 3y)^2 - 16$. - **Option 4: $(x + 3y - 4)(x + 3y + 4)$** - This option suggests a difference of squares involving $(x + 3y)^2 - 16$, which is not the correct form of the expression we derived. Thus, Option 3 is the correct choice as it accurately represents the factors of the given expression.

Question

Accepted Answer

To determine the factors of the expression $(x^2 - 6xy + 9y^2) - 16$, we need to analyze and simplify the expression step by step.

1. **Simplifying the Expression:**
   The expression $(x^2 - 6xy + 9y^2)$ is a perfect square trinomial. It can be rewritten as $(x - 3y)^2$ because:
   $$
   (x - 3y)^2 = x^2 - 2 \cdot x \cdot 3y + (3y)^2 = x^2 - 6xy + 9y^2
   $$
   Therefore, the original expression becomes:
   $$
   (x - 3y)^2 - 16
   $$

2. **Recognizing the Difference of Squares:**
   The expression $(x - 3y)^2 - 16$ is a difference of squares. It can be factored using the identity $a^2 - b^2 = (a - b)(a + b)$, where $a = (x - 3y)$ and $b = 4$.

3. **Applying the Difference of Squares Formula:**
   Using the formula, we have:
   $$
   (x - 3y)^2 - 4^2 = ((x - 3y) - 4)((x - 3y) + 4)
   $$
   Simplifying further, we get:
   $$
   (x - 3y - 4)(x - 3y + 4)
   $$

4. **Conclusion:**
   The factors of the expression $(x^2 - 6xy + 9y^2) - 16$ are $(x - 3y - 4)(x - 3y + 4)$, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1: $(x - 4)(x + 4)$**
  - This option suggests a difference of squares involving $x^2 - 16$, which does not match the original expression $(x^2 - 6xy + 9y^2) - 16$.

- **Option 2: $(x - y - 4)(x - y + 4)$**
  - This option implies a difference of squares involving $(x - y)^2 - 16$, which does not align with the expression $(x - 3y)^2 - 16$.

- **Option 4: $(x + 3y - 4)(x + 3y + 4)$**
  - This option suggests a difference of squares involving $(x + 3y)^2 - 16$, which is not the correct form of the expression we derived.

Thus, Option 3 is the correct choice as it accurately represents the factors of the given expression.