Question
Factor. \[ 16 x^{4}-80 x^{3}+100 x^{2} \] SUBMIT ANSWER
Ask by Warner Vaughan. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression factors to \(4x^{2}(2x-5)^{2}\).
Solution
Factor the expression \( 16x^4-80x^3+100x^2 \).
Factor the expression by following steps:
- step0: Factor:
\(16x^{4}-80x^{3}+100x^{2}\)
- step1: Rewrite the expression:
\(4x^{2}\times 4x^{2}-4x^{2}\times 20x+4x^{2}\times 25\)
- step2: Factor the expression:
\(4x^{2}\left(4x^{2}-20x+25\right)\)
- step3: Factor the expression:
\(4x^{2}\left(2x-5\right)^{2}\)
The factored form of the expression \(16x^{4}-80x^{3}+100x^{2}\) is \(4x^{2}(2x-5)^{2}\).
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Simplify this solution Extra Insights
To factor the expression \( 16x^{4} - 80x^{3} + 100x^{2} \), we start by observing that all the terms have a common factor of \( 4x^{2} \). Factoring out \( 4x^{2} \), we have: \[ 4x^{2}(4x^{2} - 20x + 25) \] Next, we need to factor the quadratic \( 4x^{2} - 20x + 25 \). This can be rewritten for better readability: \[ 4x^{2} - 20x + 25 = (2x - 5)^{2} \] Putting it all together, we get: \[ 4x^{2}(2x - 5)^{2} \] So the fully factored form of \( 16x^{4} - 80x^{3} + 100x^{2} \) is: \[ \boxed{4x^{2}(2x - 5)^{2}} \]
