The simplify this expression, combine like terms. Terms are like if they have the same variables and powers. To combine them, use addition and/or subtraction of the coefficients. The variables and powers do not change when you are combining.

$\displaystyle 8x$ and $\displaystyle -2x$ are like terms (both have the variable $\displaystyle x$). To combine them, you do $\displaystyle 8x - 2x = 6x$

6 and 2 are also like terms (both have no variable). To combine them, you do $\displaystyle 6 + 2 = 8$.

You now have your simplified expression: $\displaystyle 6x + 8$ which is the final answer. 

The first step is to write out $\displaystyle \small X+Y$ as   

$\displaystyle \small X+ Y= (3a^3 +2a^2 +a) + (9a^4 +3a +1)$

The next step is to re-arrange the expression so it is ordered by exponential degree, as:

$\displaystyle \small X+ Y= (9a^4 +3a^3 +2a^2 +a+3a +1)$

The final step is to sum together like terms and simplify the expression, as follows: 

$\displaystyle \small \small \small X+ Y= (9a^4 +3a^3 +2a^2 +(a+3a) +1) = (9a^4 +3a^3 +2a^2 +4a +1)$

First distribute the negative:

$\displaystyle 5x^{3}+6x-3+ \left ( -3x+2x^{2}-1 \right )$

Group like terms, meaning terms that have the same variable and same exponent:

$\displaystyle 5x^{3}+2x^{2}+6x- 3x-3 -1$

Add the like terms:

$\displaystyle 5x^{3}+2x^{2}+3x-4$

To solve you must first distribute the negative to the parentheses

$\displaystyle x^4+3x^2+14x-x^4+3x^3-2x^2$

Then you should combine like terms and you are left with

$\displaystyle 3x^3+x^2+14x$

To simplify, we will combine like terms.

$\displaystyle (2x^2 + x) + (x^2 -3x + 1)$ This has two terms with $\displaystyle x^2$, and two terms with $\displaystyle x$, so we will group those together:

$\displaystyle 2x^2 + x^2 + x - 3x + 1$ Now we can add together the like terms. Remember that if a term doesn't have a coefficient [number in front of it], there is only one of them:

$\displaystyle 3x^2 -2x + 1$

Combine the like terms (x's together and y's together).

Don't forget to distribute the negative sign through the parenthesis.

$\displaystyle (4x+7y)-(2x-5y)$

$\displaystyle =4x+7y-2x+5y$

$\displaystyle =4x-2x+7y+5y$

$\displaystyle =2x+2y$

Distribute appropriate sign and simplify:

$\displaystyle (4x^4-2x^3+x+8)-(2x^5-2x^4+3x+1)$

$\displaystyle {\color{Green} 4x^4}-2x^3+x+8-2x^5+{\color{Green} 2x^4}-3x-1$

$\displaystyle {\color{Green} 6x^4}-2x^3+{\color{Red} x}+8-2x^5-{\color{Red} 3x}-1$

$\displaystyle {\color{Green} 6x^4}-2x^3-{\color{Red} 2x}+{\color{Blue} 8}-2x^5-{\color{Blue} 1}$

$\displaystyle {\color{Green} 6x^4}-2x^3+{\color{Red} 2x}+{\color{Blue} 7}-2x^5$

Rearrange into standard form(descending degree or powers):

$\displaystyle -2x^5+6x^4-2x^3-2x+7$

$\displaystyle (3x + 7y - 5z) + (- 2x - 6y + 4z) =$

$\displaystyle 3x + 7y - 5z + - 2x - 6y + 4z =$

$\displaystyle (3x - 2x) (7y - 6y) (-5z + 4z) =$

$\displaystyle x + y - z$

When adding or subtracting polynomials with exponents, the value of the exponent never changes. The exponent does change when multiplying or dividing. 

Start by categorizing each polynomial into groups according to the variable, then according to the like variable's exponent. Because $\displaystyle 3a^{2}$ and $\displaystyle 10a^{2}$ share a common variable AND exponent, you may add the coefficients:

$\displaystyle 3a^{2} + 10 a^{2} = 13a^{2 }$

However, because $\displaystyle 5b^{2}$ does not share the same variable, you may not subtract it from $\displaystyle 13a^{2 }$. Therefore, you leave the rest as is:

$\displaystyle 13a^{2} - 5b^{2}$

To simplify this expression, it is necessary to eliminate the parentheses grouping the terms.  Distribute the negative signs.  Double negative translate the sign to a positive.

$\displaystyle -(x^2+3x)-(x^2-3x) = -x^2-3x - x^2+3x$

The answer is $\displaystyle -2x^2$.