SOLUTION: Construct a polynomial function with the following properties: third degree, only real coefficients, −1 and 3+i are two of the zeros, y-intercept is −10

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Construct a polynomial function with the following properties: third degree, only real coefficients, −1 and 3+i are two of the zeros, y-intercept is −10      Log On


   

Question 1137590: Construct a polynomial function with the following properties: third degree, only real coefficients, −1 and 3+i are two of the zeros, y-intercept is −10
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a polynomial function with the following properties:
third degree,
only real coefficients,
-1 and 3%2Bi are two of the zeros,=> there is also 3-i (complex zeros always come in pairs)
y-intercept is -10=> point (0,-10)

f%28x%29=a%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29....plug in given zeros
f%28x%29=a%28x-%28-1%29%29%28x-%283%2Bi%29%29%28x-%283-i%29%29
f%28x%29=a%28x%2B1%29%28x-3-i%29%28x-3%2Bi%29

f%28x%29=a%28x%2B1%29%28x%5E2-6x%2B9-%28-1%29%29
f%28x%29=a%28x%2B1%29%28x%5E2-6x%2B9%2B1%29
f%28x%29=a%28x%2B1%29%28x%5E2-6x%2B10%29
f%28x%29=a%28x%5E3-6x%5E2%2B10x%2Bx%5E2-6x%2B10%29
f%28x%29=a%28x%5E3-5x%5E2%2B4x%2B10%29....plug in y-intercept to calculate a

-10=a%280%5E3-5%2A0%5E2%2B4%2A0%2B10%29
-10=10a
a=-1

and, your function is
f%28x%29=-1%2A%28x%5E3-5x%5E2%2B4x%2B10%29
f%28x%29=-x%5E3%2B5x%5E2-4x-10