Remainder theorem (one method) to find first solution

Question

f(x)=x^4 -7x^3+17x^2 -15x

Reminder Theorem (one method only) to find first solution.

Synthetic division to factor polynomial

Rewrite polynomial in factored form

Find remaining solutions using appropriate method

Write polynomial in factored form

Use multiplication to return your polynomial standar form.

graph the polynomial with appropriate scale

Please help me

Answer 1

x^4 -7x^3 +17x^2 -15x

= x(x^3 -7x^2 +17x -15)

=x(x-3)(x^2 -4x +5)

most everyone stops there, but if you really want to factor further you get

= x(x-3)(x-i+.5)(x-i-.5) where i=sqr(-1)

graph it with a graphing calculator

either handheld or on line

such as Desmos

it shows a warped type of U or almost UU or W shape through the origin (0,0) which is both a y and x intercept and another x intercept x=3

above the x axis the end behavior approaches infinity, in both directions, as x approaches -inf or +infinity

the graph has no global maximums, one global & local minimum, followed by 2 inflection points

for x<0 the graph is all in quadrant II, for 0<x<3, the graph is all iin quadrant IV

for x>3, the graph is all in quadrant I

by descartes rule there're max 3 positive zeros

there's just 1, 3>1

and there're no negative zeros

there are none

4th degree polynomials have 4 zeros

in this case, 2 real, 2 imaginary

x^2 -4x +5=0

x= 4/2 +/-(1/2)sqr(16-20)

x = 1/2 +/-.5sqr-4

x = .5+/- i

x = i +1/2 or i -1/2

x = i+ .5, 0, and 3 are the 4 zeros or solutions