To determine the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3x + 2\), we need to first find the degree of each part of the expression.
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Finding the degree of \((3x^2 + 4)(x^5 - 3)\):
- The term \(3x^2\) has a degree of 2.
- The term \(x^5\) has a degree of 5.
- The degree of the product of two polynomials is the sum of their degrees, so: \[ \text{Degree of } (3x^2 + 4)(x^5 - 3) = 2 + 5 = 7. \]
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Finding the degree of \(- 12x^9\):
- The degree is clearly 9.
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Finding the degree of \(- 24x^3x\):
- Simplifying \(-24x^3x\) gives \(-24x^4\). The degree here is 4.
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Finding the degree of \(+ 2\):
- The constant term \(2\) has a degree of 0.
Now, we can compare the degrees of each part of the expression:
- The degree of \((3x^2 + 4)(x^5 - 3)\) is 7.
- The degree of \(-12x^9\) is 9.
- The degree of \(-24x^4\) is 4.
- The degree of \(+ 2\) is 0.
The highest degree among these is \(9\) from the term \(-12x^9\).
Therefore, the highest degree for the entire expression is 9.