Let's analyze and simplify the given polynomial step-by-step:
The given polynomial is:
[tex]\[5 s^6 t^2 + 6 s t^9 - 8 s^6 t^2 - 6 t^7\][/tex]
Step 1: Combine like terms
We notice that there are like terms involving [tex]\(s^6 t^2\)[/tex].
Combine the coefficients of [tex]\(s^6 t^2\)[/tex]:
[tex]\[5 s^6 t^2 - 8 s^6 t^2 = (5 - 8) s^6 t^2 = -3 s^6 t^2\][/tex]
After combining the like terms, the polynomial becomes:
[tex]\[-3 s^6 t^2 + 6 s t^9 - 6 t^7\][/tex]
Step 2: Count the number of terms
The simplified polynomial has:
[tex]\[-3 s^6 t^2\][/tex]
[tex]\[6 s t^9\][/tex]
[tex]\[-6 t^7\][/tex]
Thus, the polynomial has 3 distinct terms.
Step 3: Determine the degree of the polynomial
The degree of a term in a polynomial is the sum of the exponents of the variables within that term.
- For [tex]\(-3 s^6 t^2\)[/tex], the degree is [tex]\(6 + 2 = 8\)[/tex].
- For [tex]\(6 s t^9\)[/tex], the degree is [tex]\(1 + 9 = 10\)[/tex].
- For [tex]\(-6 t^7\)[/tex], the degree is [tex]\(7\)[/tex].
The highest degree among these terms is [tex]\(10\)[/tex].
Conclusion:
- The simplified polynomial has 3 terms.
- The degree of the polynomial is 10.
Therefore, the true statement about the polynomial after it has been fully simplified is:
It has 3 terms and a degree of 10.
