To determine the correct statement about the polynomial [tex]\( 5 s^6 t^2 + 6 s t^9 - 8 s^6 t^2 - 6 t^7 \)[/tex] after it has been fully simplified, we will perform the simplification manually and then analyze the number of terms and the degree of the polynomial.
### Step-by-Step Solution:
1. Identify the identical terms:
First, we observe that [tex]\( 5 s^6 t^2 \)[/tex] and [tex]\( -8 s^6 t^2 \)[/tex] are like terms because they contain the exact same variables raised to the same powers.
2. Combine the like terms:
To combine these like terms, we simply add or subtract their coefficients:
[tex]\[
5 s^6 t^2 - 8 s^6 t^2 = (5 - 8) s^6 t^2 = -3 s^6 t^2
\][/tex]
3. Rewrite the simplified polynomial:
Substitute the combined term back into the polynomial:
[tex]\[
-3 s^6 t^2 + 6 s t^9 - 6 t^7
\][/tex]
Now, let's analyze the resulting polynomial [tex]\( -3 s^6 t^2 + 6 s t^9 - 6 t^7 \)[/tex]:
1. Count the number of terms:
The simplified polynomial has three distinct, non-like terms:
[tex]\[
-3 s^6 t^2, \quad 6 s t^9, \quad -6 t^7
\][/tex]
So, the polynomial has 3 terms.
2. Determine the degree of the polynomial:
The degree of a term in a polynomial with multiple variables is the sum of the exponents of the variables in that term. We calculate this for each term:
- For [tex]\( -3 s^6 t^2 \)[/tex]: [tex]\( \text{degree} = 6 + 2 = 8 \)[/tex]
- For [tex]\( 6 s t^9 \)[/tex]: [tex]\( \text{degree} = 1 + 9 = 10 \)[/tex]
- For [tex]\( -6 t^7 \)[/tex]: [tex]\( \text{degree} = 0 + 7 = 7 \)[/tex]
The degree of the polynomial is the highest degree among its terms. Therefore, the degree of the polynomial is:
[tex]\[
\max(8, 10, 7) = 10
\][/tex]
### Conclusion:
After simplification, the polynomial [tex]\( -3 s^6 t^2 + 6 s t^9 - 6 t^7 \)[/tex] has 3 terms and a degree of 10.
Thus, the correct statement is:
- It has 3 terms and a degree of 10.
