SOLUTION: How do you find an nth-degree polynomial function with real coefficients satisfying the given conditions? n=3; 1 and 5i are zeros; f (-1)=-104
Algebra.Com
Question 316701: How do you find an nth-degree polynomial function with real coefficients satisfying the given conditions? n=3; 1 and 5i are zeros; f (-1)=-104
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Use the following facts:
The Fundamental Theorem of Algebra, namely that every 
-th degree polynomial function has exactly zeros, counting all multiplicities.
Complex roots ALWAYS come in conjugate pairs. That means that if 
is a zero, then 
is also a zero of the desired polynomial function.
If 
is a zero of a polynomial function in 
, then 
is a factor of the polynomial.
A family of polynomial functions of the form:
\ =\ k\left(a_0x^n\ +\ a_1x^{n-1}\ +\ \cdots\ +\ a_ix^{n-i}\ +\ \cdots\ +\ a_{n-1}x\ +\ a_n\right))
all have the same zeros.
So we know the following things:
1. The desired polynomial function has exactly 3 zeros.
2. The zeros are 
, 
, and 
.
3. The factors of the polynomial, including any possible common factor multiplier are:
(x\ -\ 5i)(x\ +\ 5i))
Multiply the factors:
\ =\ k(x\ -\ 1)(x^2\ +\ 25))
\ =\ k(x^3\ -\ x^2\ +\ 25x\ -\ 25))
Since \ =\ -104)
^3\ -\ (-1)^2\ +\ 25(-1)\ -\ 25)\ =\ k\,\cdot\,(-52))
Hence 
and
\ =\ 2x^3\ -\ 2x^2\ +\ 50x\ -\ 50)
John
RELATED QUESTIONS
Find an nth-degree polynomial function with real coefficients satisfying the given (answered by MathLover1)
Find an nth- degree polynomial function with real coefficients satisfying the given... (answered by greenestamps)
Find an nth degree polynomial function with real coefficients, a leading coefficient of 1 (answered by greenestamps,ikleyn)
find an nth degree polynomial function with real coefficients satisfying the given... (answered by drk)
Find an nth degree polynomial function with real coefficients satisfying the given... (answered by MathLover1)
Find the nth-degree polynomial function with real coefficients satisfying the given... (answered by MathLover1,tommyt3rd)
Find an nth-degree polynomial function with real coefficients satisfying the given (answered by josgarithmetic)
Find an nth degree polynomial function with real coefficients satisfying the given... (answered by Fombitz)
Find an nth degree polynomial function with real coefficients satisfying the given... (answered by josgarithmetic)