Question
Factor. \( v^{4}+2 v^{2} t^{2}+t^{4} \) \( v^{4}+2 v^{2} t^{2}+t^{4}=\square \)
Ask by Byrd Mccarthy. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( v^{4} + 2v^{2}t^{2} + t^{4} = (v^{2} + t^{2})^{2} \)
Solution
To factor the expression v⁴ + 2v²t² + t⁴, notice that it resembles the pattern of a perfect square trinomial.
Recall the formula:
(a + b)² = a² + 2ab + b²
Here, let a = v² and b = t². Then:
(v² + t²)² = (v²)² + 2(v²)(t²) + (t²)²
= v⁴ + 2v²t² + t⁴
Thus, the factored form is:
v⁴ + 2v²t² + t⁴ = (v² + t²)²
So the answer is:
(v² + t²)²
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
Did you know this expression is a perfect square trinomial? It’s true! The expression \( v^{4}+2 v^{2} t^{2}+t^{4} \) can be rewritten neatly as \( (v^{2}+t^{2})^{2} \). Recognizing patterns like these can seriously speed up your factoring game! When trying to factor expressions like this one, a common mistake is to overlook the “perfect square” aspect. Always check if you can express the trinomial as a square; it can save you time and effort. Just remember: for a perfect square trinomial, the middle term is twice the product of the square roots of the first and last terms!
