Simplifying the Polynomial: Proving That

x⁴ + (−2x³ + 2x³) + (3x² − 4x² + 3x²) + (6x − 6x) + 9 = x⁴ + 2x² + 9

Mathematics often involves simplifying expressions to reveal their true form — clean, elegant, and easy to analyze. In this article, we break down one such polynomial simplification step-by-step, showing how combining like terms leads us to the simplified expression: x⁴ + 2x² + 9

Why Simplify Polynomials?

Before diving in, it’s important to understand why simplifying polynomials matters. Simplified forms make equations easier to solve, analyze, and graph. They also reveal underlying patterns — key in algebra, calculus, and advanced math contexts.

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The Given Expression

We start with: x⁴ + (−2x³ + 2x³) + (3x² − 4x² + 3x²) + (6x − 6x) + 9

At first glance, it may seem complicated, but all terms contain like terms — parts of the polynomial that share the same variable powers.

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Step 1: Combine the x³ Terms

Look closely at the cubic (degree 3) terms: −2x³ + 2x³

These are like terms because both have x³: $$ (-2 + 2)x³ = 0x³ = 0 $$

So, these terms cancel each other out: (−2x³ + 2x³) = 0

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Step 2: Combine the x² Terms

Now examine the quadratic (degree 2) terms: 3x² − 4x² + 3x²

Group like terms: $$ (3 - 4 + 3)x² = (6 - 4)x² = 2x² $$

Thus: (3x² − 4x² + 3x²) = 2x²

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Step 3: Combine the x Terms

Look at the linear (degree 1) terms: 6x − 6x

These are also like terms: $$ (6 - 6)x = 0x = 0 $$

So, (6x − 6x) = 0

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Step 4: Keep Constant Term

Finally, the standalone constant: +9

There are no other constants to combine.

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Final Simplified Form

Now substitute all simplified components back into the original expression: $$ x⁴ + (−2x³ + 2x³) + (3x² − 4x² + 3x²) + (6x − 6x) + 9 = x⁴ + 0 + 2x² + 0 + 9 $$

Therefore: x⁴ + 2x² + 9

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Summary

By combining like terms in the polynomial — specifically:

• x³ terms: −2x³ + 2x³ = 0
• x² terms: 3x² − 4x² + 3x² = 2x²
• x terms: 6x − 6x = 0

We reduce the expression neatly to: x⁴ + 2x² + 9

This simplified form showcases the true polynomial’s core structure, helping in calculations, graphing, and deeper mathematical analysis.

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Why This Matters

• Useful in equation solving and calculus (e.g., finding derivatives or integrals)
• Reveals symmetry and simplifies function behavior
• Demonstrates the power of combining like terms

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Key takeaway: Always look for like terms—terms with identical variable parts—and combine them using arithmetic to simplify complex expressions.

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By mastering these techniques, you build a strong foundation for advanced math — remember, clarity in expressions leads to confidence in problem-solving!