How to solve polynomial functions?

Question

Solve each equation and graph the related function.

1) x^3-6x^2+10x-8=0

2) x^4+x^2-2=0

Answer 1

1)

 

By using synthetic division, we get

 

(x - 4)(x² - 2x + 2) = 0

 

 

Set the factors equal to zero.

 

x - 4 = 0          and            x² - 2x + 2 = 0

 

x = 4                                  

 

 

To solve for x from the other factor, use the quadratic formula:

 

x = (-b ± √(b² - 4ac)) / 2a

 

 

where:

a = 1

b = -2

c = 2

 

Plug in these values into the quadratic formula.  Because we are using this formula to solve for x, we will either obtain two real solutions, or two complex solutions.

 

2)

 

x⁴ + x² - 2 = 0

 

(x² + 2)(x² - 1) = 0

 

(x² + 2)(x + 1)(x - 1) = 0

 

 

Set the factors equal to zero.

 

x² + 2 = 0            ,       x + 1 = 0          ,            x + 1 = 0

 

x² = -2                         x = -1                            x = 1

 

x = ±√(-2)

 

x = -i√2

 

x = i√2

 

 

We have two real roots and two complex roots.  When you graph these polynomials, keep in mind the complex zeros are not x-intercepts.  If a real root appears an even number of times, then the polynomial touches the x-axis.  If a real root appears an odd number of times, then the polynomial crosses the x-axis. 

 

Also, when a polynomial has an even degree and leading term is positive, the values of the polynomial starts to decrease as x increases.

 

When a polynomial has an odd degree and the leading term is positive, the values of the polynomial starts to increase as x increases.

 

You will also need to know the y-intercepts.

 

The y-intercept of the first polynomial is (0, -8).

The y-intercept of the second polynomial is (0, -2).