find the real solution for each equation? find the remaining zeros of the polynomial?

Question

3x^3+4x^2−7x+2=0

 

x^4+8x^3+6x^2−5x+14=0

 

a polynomial has real coefficients. the degree is 4. two zeros are i and 9+i. 

Answer 1

First problem:

 

Use the rational root theorem and synthetic division to factor the polynomial.

 

 

2/3 | 3      4       -7          2

 

                2        4         -2

__________________________

        3      6        -3         0

 

 

You have

 

(3x - 2)(x² + 2x - 1) = 0

 

 

Set the factors equal to zero.  Use the quadratic formula for the quadratic factor to solve for x.

 

 

 

Second problem:

 

Do the same thing.

 

 

-7 |  1       8        6         -5          14

 

               -7       -7          7         -14

___________________________________

        1      1       -1          2            0

 

 

(x + 7)(x³ + x² - x + 2) = 0

 

 

 

The use it again to get a quadratic factor.

 

-2 |  1      1       -1        2

 

              -2        2       -2

________________________

       1     -1        1        0

 

 

(x + 7)(x + 2)(x² - x + 1) = 0

 

 

You know what to do here.

 

 

Last problem:

 

Use the zeros to find the polynomial.  Complex zeros have conjugates.

 

x₁ = -i

x₂ = i

x₃ = 9 - i

x₄ = 9 + i

 

 

f(x) = (x + i)(x - i)(x - [9 - i])(x - [9 + i])

 

f(x) = (x² + 1)(x²  - (9 - i)x - (9 + i)x + (81 + 1))

 

f(x) = (x² + 1)(x² - 18x + 82)