Let's subtract the two polynomials step by step.
We start with the first polynomial: \[ (6x^4y^2 - 9x^3y - 8y) \]
Now we need to subtract the second polynomial: \[ (12x^4y^2 + 3y - 12x) \]
We can rewrite the subtraction as: \[ 6x^4y^2 - 9x^3y - 8y - (12x^4y^2 + 3y - 12x) \]
Distributing the negative sign to the second polynomial, we have: \[ 6x^4y^2 - 9x^3y - 8y - 12x^4y^2 - 3y + 12x \]
Now, we can combine like terms:
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For \(x^4y^2\): \[ 6x^4y^2 - 12x^4y^2 = -6x^4y^2 \]
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For \(x^3y\): \[ -9x^3y \quad (\text{there are no } x^3y \text{ terms to combine}) \]
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For \(y\): \[ -8y - 3y = -11y \]
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For \(x\): \[ +12x \quad (\text{there are no other } x \text{ terms to combine}) \]
Putting it all together, we have: \[ -6x^4y^2 - 9x^3y + 12x - 11y \]
Now, we can determine the degree of the resulting polynomial. The degree is the highest sum of the exponents of \(x\) and \(y\) in any term.
- The term \(-6x^4y^2\) has a degree of \(4 + 2 = 6\).
- The term \(-9x^3y\) has a degree of \(3 + 1 = 4\).
- The term \(12x\) has a degree of \(1\).
- The term \(-11y\) has a degree of \(1\).
The highest degree among the terms is \(6\).
Thus, the final result of the subtraction is: \[ -6x^4y^2 - 9x^3y + 12x - 11y \]
The degree of the resulting polynomial is: \[ \text{The degree of the resulting polynomial is: } 6 \]