SOLUTION: I really need help. If you could please answer these two questions for me I would be forever grateful. Find all the roots of each equation: 1. x^4-8x^3+19x^2-32x+60 2. x^5-3x^4

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I really need help. If you could please answer these two questions for me I would be forever grateful. Find all the roots of each equation: 1. x^4-8x^3+19x^2-32x+60 2. x^5-3x^4      Log On


   

Question 709853: I really need help. If you could please answer these two questions for me I would be forever grateful.
Find all the roots of each equation:
1. x^4-8x^3+19x^2-32x+60
2. x^5-3x^4-8x^3-8x^2-9x-5
i tried to solve the 1st problem but it didnt work. for the second one i got these roots: x=-1(multiplicity of 2), 5, -i, i
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
1.
x%5E4-8x%5E3%2B19x%5E2-32x%2B60=0...easiest way is to factor this completely
write -8x%5E3 as -3x%5E3-5x%5E3, 19x%5E2 as 15x%5E2%2B4x%5E2, and -32x as -12x-20x

x%5E4-3x%5E3-5x%5E3-15x%5E2%2B4x%5E2-12x-20x-60=0..group
%28x%5E4-3x%5E3%29-%285x%5E3-15x%5E2%29%2B%284x%5E2-12x%29-%2820x-60%29=0...factor out common
x%5E3%28x-3%29-5x%5E2%28x-3%29%2B4x%28x-3%29-20%28x-3%29=0
%28x%5E3-5x%5E2%2B4x-20%29%28x-3%29=0...group
%28%28x%5E3-5x%5E2%29%2B%284x-20%29%29%28x-3%29=0
%28x%5E2%28x-5%29%2B4%28x-5%29%29%28x-3%29=0
%28%28x%5E2%2B4%29%28x-5%29%29%28x-3%29=0
%28x-5%29%28x-3%29%28x%5E2%2B4%29=0
now we can find roots
if x-5=0...=>...x=5
if x-3=0...=>...x=3
if x%5E2%2B4...=>...x%5E2=-4...=>...x=sqrt%28-4%29..=>...x=2i or x=-2i


2.
x%5E5-3x%5E4-8x%5E3-8x%5E2-9x-5=0....write -3x%5E4 as 2x%5E4-5x%5E4,-8x%5E3 as -10x%5E3%2B2x%5E2, -8x%5E2 as -10x%5E2%2B2x%5E2, and -9x as -10x%2Bx

x%5E5%2B2x%5E4%2Bx%5E3-5x%5E4-10x%5E3-5x%5E2%2Bx%5E3%2B2x%5E2%2Bx-5x%5E2-10x-5=0...group





%28x%5E4%2B2x%5E3%2Bx%5E2%2Bx%5E2%2B2x%2B1%29%28x-5%29=0
%28%28x%5E4%2Bx%5E2%29%2B%282x%5E3%2B2x%29%2B%28x%5E2%2B1%29%29%28x-5%29=0
%28x%5E2%28x%5E2%2B1%29%2B2x%28x%5E2%2B1%29%2B%28x%5E2%2B1%29%29%28x-5%29=0
%28x%5E2%2B2x%2B1%29%28x%5E2%2B1%29%29%28x-5%29=0

%28x%2B1%29%5E2%28x%5E2%2B1%29%28x-5%29=0
solutions:
if %28x%2B1%29%5E2=0...=>...x%2B1=0..=>...x=-1

if x%5E2%2B1=0...=>...x%5E2=-1..=>...x=sqrt%28-1%29...=>...x=i or x=-i
if x-5=0.....=>...x=5

+graph%28+600%2C+600%2C+-10%2C+10%2C+-300%2C+300%2C+x%5E5-3x%5E4-8x%5E3-8x%5E2-9x-5%29+