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We want to factor the following expression: x^(4)+3x^(2)y+9y^(2) Which pattern can we use to factor the expression? U and V are either constant integers or single-variable expressions. Choose 1 answer: (A) (U+V)^(2) or (U-V)^(2) (B) (U+V)(U-V) (c) We can't use any of the patterns.

We want to factor the following expression:\newlinex4+3x2y+9y2 x^{4}+3 x^{2} y+9 y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.

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Full solution

Q. We want to factor the following expression:\newlinex4+3x2y+9y2 x^{4}+3 x^{2} y+9 y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
  1. Rephrasing the given expression: Let's first rephrase the ＂Which factoring pattern can be used to factor the expression x4+3x2y+9y2x^4 + 3x^2y + 9y^2?＂
  2. Identifying the structure of the expression: Identify the structure of the given expression: x4+3x2y+9y2x^4 + 3x^2y + 9y^2. Notice that x4x^4 is a square (x2)2(x^2)^2, and 9y29y^2 is also a square (3y)2(3y)^2. The middle term, 3x2y3x^2y, is twice the product of x2x^2 and 3y3y. This suggests that the expression might be a perfect square trinomial.
  3. Recalling the pattern for a perfect square trinomial: Recall the pattern for a perfect square trinomial: (U+V)2=U2+2UV+V2(U + V)^2 = U^2 + 2UV + V^2 or (U−V)2=U2−2UV+V2(U - V)^2 = U^2 - 2UV + V^2. We need to determine if the given expression fits either of these patterns.
  4. Comparing the expression with the perfect square trinomial patterns: Compare the given expression with the perfect square trinomial patterns. We have U2=x4U^2 = x^4 (so U=x2U = x^2), V2=9y2V^2 = 9y^2 (so V=3yV = 3y), and the middle term 2UV2UV should be 2×x2×3y=6x2y2 \times x^2 \times 3y = 6x^2y. However, the middle term in our expression is 3x2y3x^2y, not 6x2y6x^2y. This means that the expression does not fit the perfect square trinomial pattern exactly.
  5. Considering the other factoring pattern: Since the expression does not fit the perfect square trinomial pattern, let's consider the other factoring pattern: (U+V)(U−V)=U2−V2(U + V)(U - V) = U^2 - V^2. This pattern is used for the difference of squares, but our expression is not a difference, it's a sum. Therefore, this pattern does not apply either.
  6. Concluding that the expression cannot be factored: Having considered the common factoring patterns and finding that none of them apply to the given expression, we conclude that the expression x4+3x2y+9y2x^4 + 3x^2y + 9y^2 cannot be factored using the patterns (U+V)2(U + V)^2, (U−V)2(U - V)^2, or (U+V)(U−V)(U + V)(U - V).

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