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?

Understanding the Context

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.

The Given Expression

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

x^4 + ( -2x^3 + 2x^3 ) + (3x^2 - 4x^2 + 3x^2) + (6x - 6x) + 9 = x^4 + 2x^2 + 9

Key Insights

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

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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Final Thoughts

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

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

Step 4: Keep Constant Term

Finally, the standalone constant:
+9