To find the polynomial [tex]\( P(x) \)[/tex] based on the given expression, we need to rewrite it clearly by combining like terms and simplifying where possible. However, each term in this expression is already distinct and cannot be further simplified with respect to one another.
Let's break down the polynomial:
1. Constant Term:
- The constant term in the polynomial is [tex]\( 3 \)[/tex].
2. Term involving [tex]\( x^7 \)[/tex]:
- This term is [tex]\( x^7 \)[/tex].
3. Term involving [tex]\( x^{a+1} \)[/tex]:
- This term is [tex]\( 5 x^{a+1} \)[/tex].
4. Term involving [tex]\( x^9 \)[/tex]:
- This term is [tex]\( -2 x^9 \)[/tex].
5. Term involving [tex]\( x^{19} \)[/tex]:
- This term is [tex]\( -x^{19} \)[/tex].
Since all the terms are distinct, the most simplified form of the polynomial [tex]\( P(x) \)[/tex] is the sum of these individual terms. Thus, we write:
[tex]\[
P(x) = 3 + x^7 + 5x^{a+1} - 2x^9 - x^{19}
\][/tex]
The final simplified polynomial [tex]\( P(x) \)[/tex] is:
[tex]\[
- x^{19} - 2 x^9 + x^7 + 5 x^{a + 1} + 3
\][/tex]
This sequence and structure show the polynomial function in its simplest form with respect to each term’s degree of [tex]\( x \)[/tex].
