= (1 - 6t + 9t^2) + (16 - 16t + 4t^2) + (25 - 30t + 9t^2) \ - Decision Point
Simplifying the Polynomial Expression: (1 - 6t + 9t²) + (16 - 16t + 4t²) + (25 - 30t + 9t²)
Simplifying the Polynomial Expression: (1 - 6t + 9t²) + (16 - 16t + 4t²) + (25 - 30t + 9t²)
Finding efficient ways to simplify complex mathematical expressions is essential in algebra—whether you're solving equations, analyzing motion, or optimizing functions. In this article, we’ll simplify the polynomial expression:
(1 - 6t + 9t²) + (16 - 16t + 4t²) + (25 - 30t + 9t²)
Understanding the Context
By combining like terms and streamlining the result, this expression becomes far easier to work with in real-world applications, including physics problems and mathematical modeling.
What Is the Given Expression?
The expression combines three quadratic polynomials:
- First: 9t² - 6t + 1
- Second: 4t² - 16t + 16
- Third: 9t² - 30t + 25
Each includes constant terms, linear (t) terms, and quadratic (t²) terms, which makes combining them straightforward once terms are grouped properly.
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Key Insights
Step-by-Step Simplification
1. Combine the Constant Terms
Add all constant components together:
1 + 16 + 25 = 42
2. Combine the Linear (t) Terms
Gather all coefficients of t:
- From first: -6t
- From second: -16t
- From third: -30t
Sum: (-6) + (-16) + (-30) = -52t
3. Combine the Quadratic (t²) Terms
Add all t² coefficients:
9t² + 4t² + 9t² = 22t²
The Simplified Expression
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Putting all components together, the fully simplified form is:
(22t² - 52t + 42)
This clean, concise polynomial is easier to analyze, differentiate, or integrate—key skills in calculus, physics, and engineering.
Why Simplify Polynomials Like This?
Simplifying expressions streamlines further calculations. For instance:
- Easier plotting and graphing
- Simplified differentiation for velocity/acceleration analysis
- Faster equation solving in applied problems
Real-World Applications
This kind of polynomial simplification appears in:
- Modeling projectile motion in physics
- Optimizing cost or profit functions in economics
- Solving problems involving quadratic relationships in geometry or engineering
Final Thoughts
Simplifying expressions like (1 - 6t + 9t²) + (16 - 16t + 4t²) + (25 - 30t + 9t²) into 22t² - 52t + 42 enhances clarity and enables faster, more accurate problem-solving. Whether you're a student, teacher, or researcher, mastering this technique strengthens your mathematical foundation for advanced topics.
Tip: Always verify simplification by re-expanding the combined expression—if terms match, your work is correct!
