To determine the highest degree of the expression given, let's break it down and work through the calculations step-by-step.
The expression is:
[tex]\[
(3 x^2+4)(x^5-3) - \frac{12 x^9-24 x}{3 x+2}
\][/tex]
1. Expand the first part:
[tex]\[
(3 x^2 + 4)(x^5 - 3)
\][/tex]
- Distribute [tex]\(3 x^2\)[/tex]:
[tex]\((3 x^2) \cdot (x^5) = 3 x^7\)[/tex]
[tex]\((3 x^2) \cdot (-3) = -9 x^2\)[/tex]
- Distribute [tex]\(4\)[/tex]:
[tex]\(4 \cdot (x^5) = 4 x^5\)[/tex]
[tex]\(4 \cdot (-3) = -12\)[/tex]
Combining these terms, we get:
[tex]\[
3 x^7 + 4 x^5 - 9 x^2 - 12
\][/tex]
2. Simplify the second part:
[tex]\[
\frac{12 x^9 - 24 x}{3 x + 2}
\][/tex]
To simplify this, note the individual terms in the numerator:
- The term [tex]\(12 x^9\)[/tex] divided by [tex]\(3 x\)[/tex] yields [tex]\(4 x^8\)[/tex].
- The term [tex]\(24 x\)[/tex] divided by [tex]\(3 x\)[/tex] yields [tex]\(8\)[/tex].
Hence, each term in the numerator must be divided by [tex]\(3 x + 2\)[/tex], resulting in two separate fractions. But note that the degree of the combined term [tex]\(\frac{12 x^9-24 x}{3 x+2}\)[/tex] will influence the overall polynomial degree.
3. Combine the simplified expressions:
When we include both parts and expand the combined expression, we see:
[tex]\[
3 x^7 + 4 x^5 - 9 x^2 - 12 - \left(\frac{12 x^9}{3 x + 2} - \frac{24 x}{3 x + 2}\)
\][/tex]
The dominant term here is [tex]\(\frac{12 x^9}{3 x + 2}\)[/tex]. As [tex]\(x\)[/tex] grows large, this term tends to behave as a degree-9 polynomial (due to the highest power in [tex]\(x\)[/tex] divided by a linear term in [tex]\(x)).
Thus, the highest degree term originating from the entire expression is \(x^9\)[/tex].
Therefore, the highest degree of the given expression is [tex]\( \boxed{9} \)[/tex].
