= (1 - 6t + 9t^2) + (16 - 16t + 4t^2) + (25 - 30t + 9t^2) \ - Treasure Valley Movers

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)

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.

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

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!