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We want to factor the following expression: (x-3)^(2)-10(x-3)+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:\newline(x−3)2−10(x−3)+25 (x-3)^{2}-10(x-3)+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:\newline(x−3)2−10(x−3)+25 (x-3)^{2}-10(x-3)+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. Expression Analysis: We are given the expression (x−3)2−10(x−3)+25(x-3)^{2}-10(x-3)+25 and asked to identify a factoring pattern that can be used to factor it. Let's first rewrite the expression to see if it resembles any common factoring patterns.
  2. Perfect Square Trinomial: The given expression is a quadratic in the form of a perfect square trinomial. A perfect square trinomial is an expression that can be written as A+BA+B^22 or A−BA-B^22, where AA and BB are expressions. The standard form of a perfect square trinomial is A2±2AB+B2A^2 \pm 2AB + B^2.
  3. Comparison with Standard Form: Let's compare the given expression with the standard form of a perfect square trinomial. We have A=(x−3)A = (x-3) and we need to find BB such that the expression matches A2−2AB+B2A^2 - 2AB + B^2. We can see that A2A^2 is (x−3)2(x-3)^2, and B2B^2 should be 2525, which means BB is 55 since 52=255^2 = 25.
  4. Matching the Middle Term: Now, we need to check if the middle term −10(x−3)-10(x-3) matches the −2AB-2AB term in the perfect square trinomial pattern. For our expression, AA is (x−3)(x-3) and BB is 55, so −2AB-2AB would be −2⋅(x−3)⋅5-2\cdot(x-3)\cdot5, which simplifies to −10(x−3)-10(x-3).
  5. Factoring Using the Pattern: Since the given expression matches the pattern A2−2AB+B2A^2 - 2AB + B^2, where AA is (x−3)(x-3) and BB is 55, we can factor it using the pattern (A−B)2(A-B)^2. Therefore, the factored form of the expression is ((x−3)−5)2((x-3)-5)^2 or (x−3−5)2(x-3-5)^2, which simplifies to (x−8)2(x-8)^2.
  6. Successful Factoring: We have successfully factored the expression using the pattern (A−B)2(A-B)^2. This corresponds to choice AA from the given options, which are (U+V)2(U+V)^2 or (U−V)2(U-V)^2.

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