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Question 143384This question is from textbook Mathmatical Analysis
: The problem is x(x+1)squared(x+2)=0; 0, -1, 2.
Sorry I dont know how to make an exponent on the computer. I am supposed to use the 0, -1, and 2. to determine by substitution which of the given numbers, if any satisfy the equation. So far I have getting that -1 and 0 work! I have a question about my work though. Am I supposed to distribute the x first or foil first? Thanks so much for your time! Tarah
This question is from textbook Mathmatical Analysis
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! It doesn't matter whether you distribute the x first or use the FOIL process first -- remember the commutative property of multiplication? From a practical point of view, you will find it easier to do the FOIL thing first and then distribute the x because distributing the x is just a matter of going across the polynomial that resulted from the FOIL process and increasing the exponents on the variables by 1.
Having said all of that, you really don't need to use FOIL or distribute anything. Just remember the Zero Product Rule:  if and only if  or  .
Applying the zero product rule to  you get:
 , or
 =>  , or
=> , (because there are 2 factors of  ) or
 => 
And those are all of the roots of the given equation. We are sure of this fact because we know that if you did use FOIL and distribution to multiply the factors, you would end up with a polynomial equation with  as the high order term, making it a 4th degree polynomial equation. The Fundamental Theorem of Algebra tells us that a polynomial equation of degree n has exactly n roots (although some or all of them may be complex numbers). You have a 4th degree equation, and 4 roots, and all is right with the world.
Notice that -2 is a root, but 2 is not.
And one last thing: If you want to indicate an exponent on the computer, just use the caret mark (Shift-6), like this: x(x + 1)^2(x + 2).
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