To subtract the polynomials \((x^3 + 6xy + 3y^2) - (4x^3 + 9xy + y^2)\), we distribute the negative sign to the second polynomial and then combine like terms.
Here's the subtraction step by step:
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Rewrite the expression: \[ x^3 + 6xy + 3y^2 - (4x^3 + 9xy + y^2) \]
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Distribute the negative sign: \[ x^3 + 6xy + 3y^2 - 4x^3 - 9xy - y^2 \]
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Combine like terms:
- For \(x^3\): \(x^3 - 4x^3 = -3x^3\)
- For \(xy\): \(6xy - 9xy = -3xy\)
- For \(y^2\): \(3y^2 - y^2 = 2y^2\)
Putting it all together, we have: \[ -3x^3 - 3xy + 2y^2 \]
Now, let's determine the degree of the resulting polynomial. The degree is the highest power of \(x\) or \(y\) in the polynomial.
- The term \(-3x^3\) has a degree of 3.
- The term \(-3xy\) has a degree of \(1 + 1 = 2\).
- The term \(2y^2\) has a degree of 2.
The highest degree is 3, so the degree of the resulting polynomial is 3.
Final Answer: \[ -3x^3 - 3xy + 2y^2 \] Degree: 3