When solving a polynomial you must combine like terms. Like terms have the same degree value. A degree value is the number of exponents a term has.

\(\displaystyle 6x^2 + 2y + 4x + 3y - 2x =\)

\(\displaystyle (6x^2) + (2y + 3y) + (4x - 2x) =\)

\(\displaystyle 6x^2 + 5y + 2x\)

When adding polynomials, we simply look at each term and combine like terms. Like terms in this case are terms that have the same variables. 

\(\displaystyle (3x^2 -4x +2) + (2x^2 +9x-1)\)

We can remove the parentheses in the problem, because doing so will not change the order of operations. So, we get

\(\displaystyle 3x^2 -4x +2 + 2x^2 +9x-1\)

Now, we find like terms.

\(\displaystyle {\color{Red} 3x^2} {\color{Green} -4x} {\color{Blue} +2} {\color{Red}+ 2x^2} {\color{Green} +9x}{\color{Blue} -1}\)

You can see all like terms have been highlighted in different colors. Now, we can combine them.

\(\displaystyle {\color{Red} 3x^2 + 2x^2} = 5x^2\)

\(\displaystyle {\color{Green} -4x + 9x} = 5x\)

\(\displaystyle {\color{Blue} 2-1} = 1\)

Note that we only add the coefficients together. Ther variables themselves do not change.

Writing that together, we get 

\(\displaystyle 5x^2 + 5x +1\)

To solve this problem, we will group and combine like terms.  We can remove the parentheses because they do not change the answer. We get

\(\displaystyle (2x+4y) + (6x-2y)\)

\(\displaystyle 2x + 4y + 6x - 2y\)

\(\displaystyle 2x + 6x + 4y - 2y\)

\(\displaystyle 8x + 2y\)

Note that when you combine like terms, the variables do not change.  We only combine the coefficients. 

When adding polynomials, we can remove the parentheses because this does not have an effect on the answer. So,

\(\displaystyle (4x^2 - 2x - 4) + (9x^2 - 3x +1)\)

becomes

\(\displaystyle 4x^2 - 2x - 4 + 9x^2 - 3x +1\)

Now, we combine like terms.  Like terms are considered terms with the same variables.  So,

\(\displaystyle {\color{Red} 4x^2} {\color{Green} - 2x} {\color{Blue} - 4} {\color{Red} + 9x^2} {\color{Green} - 3x} {\color{Blue} +1}\)

where each different colored terms are like terms.  We combine them and get

\(\displaystyle {\color{Red} 13x^2} {\color{Green} - 5x} {\color{Blue} - 3}\)

Note that we do not change the variables when we add them, we only combine the coefficients.

Simplify the following expression:

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

Let's begin by putting like terms next to eachother:

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

Next, we keep the exponents the same (because we are only adding/subtracting), but we treat the numbers out in front just like regular subtraction or addition.

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

So our answer is

\(\displaystyle 2x^3-5x^2\)

For this problem you need to combine your like terms.

\(\displaystyle 4x^3-x^3+x^2+44=x^3\)  simplifies to  \(\displaystyle 3x^3+x^2+44=x^3\)

But you must bring the other \(\displaystyle x^3\) over by subtraction so:

\(\displaystyle 2x^3+x^2+44=0\)

When finding our answer the first step is to get rid of the parentheses symbol so we can combine like terms, so we start with the following equation:

\(\displaystyle (4p^{2}-4m)+(6p^{3}+6m-7v)\)

Then we have

\(\displaystyle 4p^{2}-4m+6p^{3}+6m-7v\)

We simple add like terms. Remember terms with the same power and variables can be added. So we have

\(\displaystyle 6p^{3}+4p^{2}+6m-4m-7v\)

The only term that will change from our original problem is the one linked to the variable m.

\(\displaystyle 6m-4m=2m\)

So then we have 

\(\displaystyle 6p^{3}+4p^{2}+2m-7v\)

When subtracting polynomials, we must first distribute the negative through the second polynomial.

\(\displaystyle (-2x^2 + 4x - 9) {\color{Red} - }(-5x^2 - 2x +1)\)

To do this, we simply take each term in the second polynomial times -1.  We get

\(\displaystyle (-2x^2 + 4x - 9) +5x^2+2x-1\)

We can remove the parentheses from the first polynomial.

\(\displaystyle -2x^2 + 4x - 9 +5x^2+2x-1\)

Now, we combine like terms.  Like terms are the terms that have the same variables.  

\(\displaystyle {\color{Red} -2x^2} {\color{Green} + 4x} {\color{Blue} - 9} {\color{Red} +5x^2}{\color{Green} +2x}{\color{Blue} -1}\)

The like terms have been highlighted.  When we combine them, we get

\(\displaystyle {\color{Red} 3x^2} {\color{Green} + 6x} {\color{Blue} -10}\)

Multiply making sure to distribute the negative sign:

\(\displaystyle -2x(x^4-x^2+4)=(-2x)(x^4)-(-2x)(x^2)+(-2x)(4)\)

\(\displaystyle =-2x^5+2x^3-8x\)

\(\displaystyle (x^2+2)(-x^3+3x-1)=-x^5+3x^3-x^2-2x^3+6x-2\)

\(\displaystyle =-x^5+x^3-x^2+6x-2\)