what is the highest degree for the expression (3x2+4)(x5−3)−12x9−24x3x+2 ?

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Answered by GPT-4o mini
To find the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3 + 2\), we first need to determine the degree of each part of the expression.

1. **First, find the degrees of the products:**
- The term \(3x^2 + 4\) has a degree of \(2\) (the highest exponent is \(2\) from \(3x^2\)).
- The term \(x^5 - 3\) has a degree of \(5\) (the highest exponent is \(5\) from \(x^5\)).
- The product \((3x^2 + 4)(x^5 - 3)\) has a degree of \(2 + 5 = 7\) (the degree is the sum of the degrees of the individual polynomials).

2. **Next, find the degrees of the remaining terms:**
- The term \(-12x^9\) has a degree of \(9\).
- The term \(-24x^3\) has a degree of \(3\).
- The constant term \(+2\) has a degree of \(0\) (constants are considered degree \(0\)).

3. **Now, combine all the information:**
- The highest degree from the product \((3x^2 + 4)(x^5 - 3)\) is \(7\).
- The term \(-12x^9\) has the highest degree of \(9\).

4. **Conclusion:**
The highest degree in the entire expression is from the term \(-12x^9\), which is \(9\).

Thus, the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3 + 2\) is \(9\).

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