Find all rational zeros of the polynomials

Question

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = x⁵ + 3x⁴ − 25x³ − 79x² + 100

x

=

Write the polynomial in factored form.

P(x)

=

Answer 1

Paul gave a great suggestion. With this particular problem, the plot reveals 4 roots (hint: plot from -5 to 5; a glance at the function shows that no roots can have absolute value 10 or above). The shape of the function shows one multiple root.

If you are able to create the function, it is also easy to implement Mark's solution (you won't have to make 18 separate calculations by hand, just create a list of 18 values to calculate with the function; this is easy to do in Excel). The advantage of the plot is that you identify the multiple. Since there is only one extra root, you can actually easily determine it without plotting, as you can multiply the four roots and know that the fifth one will be -100/the product of the other four, as the product of the five roots is -100.

Answer 2

Yes, Mark, the rational root theorem works...but with a constant term of 100, there are a lot of factors to check.

I suggest using a graphing calculator to get an idea about the roots first.