Question

1. which expression is equivalent to ((x - xy^2)(yz^2 + 2x))?

○ (xyz^2 - xy^3z^2 + 2x^2 + 2x^2y^2)
○ (xyz^2 - xy^3z^2 + 2x^2 - x^2y^2)
○ (xyz^2 - xy^2z^2 + 2x - 2x^2y^2)
○ (xyz^2 - xy^3z^2 + 2x^2 - 2x^2y^2)

Explanation:

Step1: Apply the distributive property (FOIL method)

Multiply each term in the first binomial \((x - xy^{2})\) by each term in the second binomial \((yz^{2}+2x)\):

• Multiply \(x\) by \(yz^{2}\): \(x\times yz^{2}=xyz^{2}\)
• Multiply \(x\) by \(2x\): \(x\times2x = 2x^{2}\)
• Multiply \(-xy^{2}\) by \(yz^{2}\): \(-xy^{2}\times yz^{2}=-xy^{3}z^{2}\)
• Multiply \(-xy^{2}\) by \(2x\): \(-xy^{2}\times2x=-2x^{2}y^{2}\)

Step2: Combine the terms

After multiplying, we combine the resulting terms:
\(xyz^{2}+2x^{2}-xy^{3}z^{2}-2x^{2}y^{2}\)
Rearranging the terms in the order of the options, we get:
\(xyz^{2}-xy^{3}z^{2}+2x^{2}-2x^{2}y^{2}\)

Answer:

The fourth option: \( xyz^{2}-xy^{3}z^{2}+2x^{2}-2x^{2}y^{2} \)