To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle a(a + b) - b(a+b)$

$\displaystyle a^2 + ab - ab - b^2$

$\displaystyle a^2 -b^2$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle a(x + y) - a(x-y)$

$\displaystyle ax + ay - ax + ay$

$\displaystyle 2ay$

Simplify the following expression:

$\displaystyle (9x^8+4x^4-5x+4)-(-x^8+5x^5+4x^4+15x+3)$

To simplify the expression, we first need to distribute the negative sign.

$\displaystyle (9x^8+4x^4-5x+4)+x^8-5x^5-4x^4-15x-3)$

Next, remove the other parentheses, and rearrange the terms to get similar exponents next to eachother:

$\displaystyle 9x^8+x^8-5x^5+4x^4-4x^4-5x-15x+4-3$

Finally, combine each set of like terms and you will have your answer:

$\displaystyle 10x^8-5x^5-20x+1$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle -x^2 -5xy$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle -a-4b$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle -6x - 2y +3$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle -3m + 3n$

Subtracting polynomials is very simple. Once we've changed the sign for the polynomial right of the subtraction sign, the problem becomes a matter of collecting like terms.

$\displaystyle (x^2+2x+3)-(x^2-7x+2)$

$\displaystyle x^2+2x+3-x^2+7x-2$

we've changed the sign of every term of the polynomial right of the subtraction sign because we've distributed the subtraction sign to get rid of the parentheses.

Now we can collect like terms to solve for the final answer. 

$\displaystyle x^2{\color{Red} -x^2}+2x{\color{Red} +7x}+3{\color{Red} -2}$

$\displaystyle {\color{Blue} 9x+1}$

Subtracting polynomials is very simple. Once we've changed the sign for the polynomial right of the subtraction sign, the problem becomes a matter of collecting like terms.

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

$\displaystyle x^2+x+9{\color{Red} -x^3+x^2-4}$

we've changed the sign of every term of the polynomial right of the subtraction sign because we've distributed the subtraction sign to get rid of the parentheses.

Now we can collect like terms to solve for the final answer. 

$\displaystyle {\color{Red} -x^3}+x^2{\color{Red} +x^2}+x+9{\color{Red} -4}$

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

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle -2x -2z -mx -mz$