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We want to factor the following expression: x^(4)+10 x+25 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+10x+25 x^{4}+10 x+25 \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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Q. We want to factor the following expression:\newlinex4+10x+25 x^{4}+10 x+25 \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. Identify Structure: Identify the structure of the given expression.\newlineThe given expression is x4+10x+25x^4 + 10x + 25. We need to determine if this expression fits any known factoring patterns.
  2. Check for Patterns: Check for common factoring patterns.\newlineThe common factoring patterns for quadratic-like expressions are:\newline(A) (U+V)2=U2+2UV+V2(U + V)^2 = U^2 + 2UV + V^2\newline(B) (U−V)2=U2−2UV+V2(U - V)^2 = U^2 - 2UV + V^2\newline(C) (U+V)(U−V)=U2−V2(U + V)(U - V) = U^2 - V^2\newlineWe need to see if the given expression fits any of these patterns.
  3. Compare with Patterns: Compare the given expression with the patterns.\newlineThe given expression x4+10x+25x^4 + 10x + 25 does not immediately resemble any of the patterns because it is not a quadratic expression (it has a term with x4x^4). However, we can consider x4x^4 as (x2)2(x^2)^2 and 2525 as 525^2. This suggests we might be able to use pattern (A) or (B) if the middle term fits.
  4. Attempt Pattern (A): Attempt to fit the expression into pattern (A).\newlineLet's see if the expression fits the pattern (A) (U+V)2=U2+2UV+V2(U + V)^2 = U^2 + 2UV + V^2. We have U2=x4U^2 = x^4 (so U=x2U = x^2) and V2=25V^2 = 25 (so V=5V = 5). The middle term for (U+V)2(U + V)^2 would be 2UV=2(x2)(5)=10x22UV = 2(x^2)(5) = 10x^2, which does not match the middle term of the given expression, which is 10x10x.
  5. Conclude No Factoring: Conclude that the expression cannot be factored using the given patterns.\newlineSince the middle term of the given expression does not match the middle term that would result from using pattern (A)(A) or (B)(B), and pattern (C)(C) is not applicable to this expression, we conclude that the expression x4+10x+25x^4 + 10x + 25 cannot be factored using any of the given patterns.

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