#### Andy's, a 9th grader, Discovery: Finding the equation, given its roots

Beginning Variables:
@ is the number of roots
y=0
Sections 1,2, and 4 must only have 1 term, Section 3 may have any number of terms

The steps and instructions in each section show how to get each term
Section 1:
1:X^@

Section 2:
1:(Take the opposite of the sum of the roots) * X^@-1

Section 3:
1: number of terms here is equal to (@-2)
2: (Take the sum of the roots multiplied (2+y) at a time )*X^(@-2-y)
3: if y is odd, then take the opposite of the number you got in step 2
4: Increase the value of y by 1 (y=y+1)
5 if @-2-y > 0 then go to step 2, elsewise go to the next section

Section 4:
1:Take the product of the roots
2: If @ is odd, take the opposite of the answer in step 1
Add the terms you got in each section together in order, and a "=" sign, and put 0 on the other side of the =

Example
The words not in the { } are the actual terms. The words inside the { } shows the work that got the terms
Suppose the roots are 3 , 4 , and 5. (note that there can be any number of roots, and they dont have to be in any specific order)

Since the number of roots is 3, @=3
Section 1: The first term is x^3 {x^@, and since @=3 the term is x^3 }

Section 2: The second term is -12x^2 {The sum of the roots is 12 (3+4+5) and the opposite of that is -12. then take -12 times x^@-1, so this would be -12 times x^2, so the term is -12x^2}

Section 3: The 3rd though @-2 terms is (realize this has to be somewhere between 1 term and however many roots you have minus 2) 47x {The sum of the terms taken 2 at a time (2+y roots at a time, and since y is zero now the sum would be (3*4)+(4*5+(3*5) times x^1 (x*(@-2-y), and y=0 and @=3 then it is x^1, and since y is not odd, we dont take the opposite. then the next step is to increase y by 1, so y now equals 1 (y=y+1 and y began as zero), and the last step in this section is to see if @-2-y isnt greater then 0( 3-2-1 isnt greater than 0 ) we go on to to section 4}

Section 4:The last term is -60 {we take the products of the roots (3*4*5) is equal to 60, and since @=3 and 3 is odd, we take the opposite of 60, which is -60}

The last step is to combine all the the terms in order and set =0 , so the final equation is
x^3-12x^2+47x-60=0

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