Answer :
We start with the expression
$$-21x^5 + 30x^6 + 21x^6.$$
**Step 1. Identify like terms.**
Notice that the terms $30x^6$ and $21x^6$ have the same variable part $x^6$. These can be combined by adding their coefficients.
**Step 2. Combine like terms.**
Add the coefficients of $x^6$:
$$30 + 21 = 51.$$
Thus, the two terms become
$$51x^6.$$
So the expression now is
$$-21x^5 + 51x^6.$$
**Step 3. Write the simplified expression.**
The simplified expression is
$$-21x^5 + 51x^6.$$
**Step 4. (Optional Factoring)**
If we want to factor the expression, notice that both terms contain a common factor of $x^5$. Factoring $x^5$ out, we get
$$x^5(-21 + 51x),$$
which can also be written as
$$x^5(51x - 21).$$
**Intermediate Result: Sum of Coefficients for $x^6$.**
The sum of the coefficients for the $x^6$ terms is $51$.
Thus, the final results are:
- The sum of the coefficients for $x^6$ is $51$.
- The simplified expression is $$-21x^5 + 51x^6$$ or equivalently $$x^5(51x - 21).$$
$$-21x^5 + 30x^6 + 21x^6.$$
**Step 1. Identify like terms.**
Notice that the terms $30x^6$ and $21x^6$ have the same variable part $x^6$. These can be combined by adding their coefficients.
**Step 2. Combine like terms.**
Add the coefficients of $x^6$:
$$30 + 21 = 51.$$
Thus, the two terms become
$$51x^6.$$
So the expression now is
$$-21x^5 + 51x^6.$$
**Step 3. Write the simplified expression.**
The simplified expression is
$$-21x^5 + 51x^6.$$
**Step 4. (Optional Factoring)**
If we want to factor the expression, notice that both terms contain a common factor of $x^5$. Factoring $x^5$ out, we get
$$x^5(-21 + 51x),$$
which can also be written as
$$x^5(51x - 21).$$
**Intermediate Result: Sum of Coefficients for $x^6$.**
The sum of the coefficients for the $x^6$ terms is $51$.
Thus, the final results are:
- The sum of the coefficients for $x^6$ is $51$.
- The simplified expression is $$-21x^5 + 51x^6$$ or equivalently $$x^5(51x - 21).$$
