11. Algebraic Expressions
For solving problems in the GMAT and the GRE, you need to memorize certain formule that will help you factor and expand algebraic expressions.
1. $(x+a)(x+b)=x^2+(a+b)x+ab$
2. $(x+a)^2=x^2+2ax+a^2$
3. $(x-a)^2=x^2-2ax+a^2$
4. $(x+a)(x-a)=x^2-a^2$
5. $(x+y)^3=x^3+3xy(x+y)+y^3=x^3+3x^2y+3xy^2+y^3$
6. $(1+a)(1+b)(1+c)=1+a+b+c+ab+bc+ca+abc$
7. $a^3+b^3=(a+b)(a^2-ab+b^2)$
8. $a^3-b^3=(a-b)(a^2+ab+b^2)$

It is not just important to know these formule, but also how to use them.
While we cannot anticipate every possible application that might arise, we can study some examples to understand better how these are to be used.
1. factorize: $x^2+5x+6$. Simple. We notice 5=2+3, and 6=2*3, so $x^2+5x+6=(x+3)(x+2)$
2. solve: $x^2+5x+6=0$. Again, $x^2+5x+6=(x+3)(x+2)=0$. This means $(x+3)=0$ or $(x+2)=0$, yielding roots as $x=-3$ or $x=-2$
3. simplify: $({x^100-y^100})/({x^50-y^50})$. We note the expression equals: $({(x^50+y^50)(x^50-y^50)})/(x^50-y^50)=(x^50+y^50)$

To be able to solve problems involving algebraic expressions in a timely way, it is best to memorize the formule above.
Sure, you can multiply them out in the exam if you wish, but remember, you are competing against many others who would have these memorized.


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