To determine the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3 x + 2\), we first need to find the degree of each part of the expression.
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For the product \((3x^2 + 4)(x^5 - 3)\):
- The degree of \(3x^2\) is 2.
- The degree of \(x^5\) is 5.
- When multiplying, the degrees add, so the degree of \((3x^2 + 4)(x^5 - 3)\) is \(2 + 5 = 7\).
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Next, consider \(-12x^9\):
- This term has a degree of 9.
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Now, look at \(-24x^3x\):
- This can be simplified to \(-24x^4\), which has a degree of 4.
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Finally, the constant term \(+2\):
- This term has a degree of 0.
Now we compile the degrees from each part:
- From \((3x^2 + 4)(x^5 - 3)\): degree 7
- From \(-12x^9\): degree 9
- From \(-24x^4\): degree 4
- From \(+2\): degree 0
The highest degree among these terms is \(\text{degree } 9\) from \(-12x^9\).
Thus, the highest degree for the expression is:
9