SOLUTION: Hello! Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions. P has degree 2, and zeros 1 + i and 1 - i. thanks

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hello! Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions. P has degree 2, and zeros 1 + i and 1 - i. thanks      Log On


   

Question 199782: Hello!
Find a polynomial with integer coefficients and a leading coefficient of one that satisfies the given conditions.
P has degree 2, and zeros 1 + i and 1 - i.
thanks for the homework help!
Found 2 solutions by solver91311, jim_thompson5910:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


If and are roots of the equation , then and are factors of the trinomial .

So the product of:



will result in the desired monic univariate polynomial.

Hints:

1. Treat and as single numbers and apply FOIL.

2. results in the difference of two squares.

3. Remember that

John

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x=1%2Bi or x=1-i Start with the given zeros


x-1=i or x-1=-i Get the constants to the left side


%28x-1%29%5E2=%28i%29%5E2 or %28x-1%29%5E2=%28-i%29%5E2 Square both sides


%28x-1%29%5E2=-1 or %28x-1%29%5E2=-1 Square i and -i to get -1


Since the equations are the same, we can focus on the first equation %28x-1%29%5E2=-1


x%5E2-2x%2B1=-1 FOIL


x%5E2-2x%2B1%2B1=0 Add 1 to both sides.


x%5E2-2x%2B2=0 Combine like terms.


So the polynomial f%28x%29=x%5E2-2x%2B2 has zeros of 1+i and 1-i



-------------------------------


Check:

You can check the answer with the quadratic formula.


Notice that the quadratic x%5E2-2x%2B2 is in the form of Ax%5E2%2BBx%2BC where A=1, B=-2, and C=2


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%282%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-2, and C=2


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%282%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%281%29%282%29+%29%29%2F%282%281%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4-8+%29%29%2F%282%281%29%29 Multiply 4%281%29%282%29 to get 8


x+=+%282+%2B-+sqrt%28+-4+%29%29%2F%282%281%29%29 Subtract 8 from 4 to get -4


x+=+%282+%2B-+sqrt%28+-4+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+2%2Ai%29%2F%282%29 Take the square root of -4 to get 2%2Ai.


x+=+%282+%2B+2%2Ai%29%2F%282%29 or x+=+%282+-+2%2Ai%29%2F%282%29 Break up the expression.


x+=+%282%29%2F%282%29+%2B+%282%2Ai%29%2F%282%29 or x+=++%282%29%2F%282%29+-+%282%2Ai%29%2F%282%29 Break up the fraction for each case.


x+=+1%2Bi or x+=++1-i Reduce.



So the zeros are x+=+1%2Bi or x+=+1-i which confirms our answer.