How do you factor 2x^5-7x^4-16x^3+5x^2-4x+20 and graph it? Given zeros are x= -2 x= 5 and x= 1.

Question

Hi, I'd like a step by step explanation of how to factor this polynomial equation and graph it.

Answer 1

A very useful site for viewing the graph of higher degree polynomials is

www.wolframalpha,com

 

An app for smartphone is available for a very small fee.

Answer 2

We are given three zeros.  We can convert the given zeros into factor form.

 

(x + 2)(x - 5)(x - 1)

 

 

Next, we expand the factors.

 

(x + 2)(x² - 6x + 5) =

 

x³ - 6x² + 5x + 2x² - 12x + 10 =

 

x³ - 4x² - 7x + 10

 

 

Next, we can divide the original polynomial by this new factor using long division.

 

                   

                                                            2x² + x  + 2

                              _____________________________

(x³ - 4x² - 7x + 10) | 2x⁵ - 7x⁴ - 16x³ + 5x²  - 4x + 20

                               2x⁵ - 8x⁴  - 14x³ + 20x²

                               _____________________

                                         x⁴  - 2x³  - 15x² - 4x

                                         x⁴  - 4x³ -   7x² + 10x

                                         ______________________

                                                2x³  - 8x²   - 14x  + 20

                                                2x³  - 8x²  - 14x  + 20

                                                _____________________

                                                                           0

 

Since the remainder is zero, we have our other factor.

 

The complete factoring of the polynomial is    (x + 2)(x - 5)(x - 1)(2x² + x + 2)

 

Note that the factor we found using long division is not factorable.  By examining this factor, the zeros that result from it are complex.  Using the quadratic formula will prove this.

 

 

Now to graph. 

 

Since our polynomial is positive with a negative degree, the graph starts out with increasing, and finishes increasing.

 

Start your graph increasing in a straight line from the interval (-∞, -2).  Cross the x-axis at (-2, 0).  Then keep increasing in a straight line until you reach the point (0, 20).  After that point, decrease in a straight line and cross the x-axis at the point (1, 0).  Your minimum value will be located between the point (1, 0) and (5, 0).  When you get to the minimum point, increase back up and cross the point (5, 0).  Then finish up increasing in the interval (5, ∞).