SOLUTION: Suppose x^2 + bx + c has zeros r1 and r2, what is the polynomial in factored form multiplying the factored form out. What are then the relationships between this product and the

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Suppose x^2 + bx + c has zeros r1 and r2, what is the polynomial in factored form multiplying the factored form out. What are then the relationships between this product and the       Log On


   

Question 250263: Suppose x^2 + bx + c has zeros r1 and r2, what is the polynomial in factored form multiplying the factored form out. What are then the relationships between this product and the coefficients in the original polynomial?
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
Let's start with original equation:
x%5E2+%2B+bx+%2B+c
We set it equal to 0 and complete the square.
rewrite the expression as:
x%5E2+%2B+bx+%2B+_____+%2B+c++=+0+%2B++_____
The blanks are waiting for certain numbers.
step 1 - take 1/2 middle term and square it. Place that answer in the first blank and the opposite in the second blank as:
x%5E2+%2B+bx+%2B+%28b%2F2%29%5E2+%2B+c+=+0+%2B+%28b%2F2%29%5E2+
step 2 - subtract c from both sides to get
x%5E2+%2B+bx+%2B+%28b%2F2%29%5E2+=+-c+%2B+%28b%2F2%29%5E2+
step 3 - express the left side as a quantity squared as
%28x+%2B+%28b%2F2%29%29%5E2+=+-c+%2B+%28b%2F2%29%5E2+
Step 4 - simplify the right side as
%28x+%2B+%28b%2F2%29%29%5E2+=+%28b%5E2-4c%29%2F4+
step 5 - take a square root of both sides to get
x+%2B+%28b%2F2%29+=+%2B-+sqrt%28%28b%5E2-4c%29%2F4%29+
step 6 - add b/2 to both sides to get
x+=+-b%2F2+%2B-+%28sqrt%28b%5E2-4c%29%29%2F2+
We now have our 2 roots as
r1 = x+=+-b%2F2+%2B+sqrt%28%28b%5E2-4c%29%29%2F2+
and
r2 = x+=+-b%2F2+-+sqrt%28%28-b%5E2-4c%29%29%2F2+.
r1 + r2 = -b in this case, but in general it will be -b/2a.
r1*r2 = c in this case and in general it is c.