x^4 + ( -2x^3 + 2x^3 ) + (3x^2 - 4x^2 + 3x^2) + (6x - 6x) + 9 = x^4 + 2x^2 + 9 - Richter Guitar
Simplifying the Polynomial: Proving That
x⁴ + (−2x³ + 2x³) + (3x² − 4x² + 3x²) + (6x − 6x) + 9 = x⁴ + 2x² + 9
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
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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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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
