Questions: Factor the following. [ y^2-x^3 y^2 ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. ( y^2-x^3 y^2 = ) B. The polynomial is prime.

Solution Steps

To factor the given polynomial \( y^{2} - x^{3} y^{2} \), we can look for common factors in each term. Notice that both terms have a common factor of \( y^{2} \). We can factor out \( y^{2} \) from the polynomial.

1. Identify the common factor in both terms.
2. Factor out the common factor.

We start with the polynomial given by \[ y^{2} - x^{3} y^{2}. \]

Both terms in the polynomial share a common factor of \( y^{2} \). We can factor this out: \[ y^{2} (1 - x^{3}). \]

The expression \( 1 - x^{3} \) can be factored using the difference of cubes formula, which states that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, we can rewrite \( 1 \) as \( 1^3 \): \[ 1 - x^{3} = (1 - x)(1^2 + 1 \cdot x + x^2) = (1 - x)(1 + x + x^{2}). \]

Combining the factors, we have: \[ y^{2} (1 - x)(1 + x + x^{2}). \]

Final Answer