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

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