SOLUTION: I need all the roots and zeros to these problems x^3+3x^2-5x-4=0 x^4-2x^3-7x^2+18x-18+=0

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Question 107254: I need all the roots and zeros to these problems
x^3+3x^2-5x-4=0
x^4-2x^3-7x^2+18x-18+=0
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
First example.
A quick graph of the function shows that x=-4 looks like a root of the equation.
+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E3%2B3x%5E2-5x-4%29+
f%28x%29=x%5E3%2B3x%5E2-5x-4
f%284%29=%28-4%29%5E3%2B3%2A%28-4%29%5E2-5%2A%28-4%29-4
f%284%29=-64%2B48%2B20-4
f%284%29=0
You can then find the polynomial g(x) such that
f%28x%29=g%28x%29%2A%28x%2B4%29
where g%28x%29=ax%5E2%2Bbx%2Bc and
f%28x%29=%28ax%5E2%2Bbx%2Bc%29%2A%28x%2B4%29
f%28x%29=ax%5E3%2B%28b%2B4a%29x%5E2%2B%28c%2B4b%29%2Ax%2B4c
Compare that to your original equation.
f%28x%29=x%5E3%2B3x%5E2-5x-4 and you see that
1.a=1
2.b%2B4a=3
3.c%2B4b=-5
4.4c=-4
From 1 and 2,
b%2B4=3
b=-1
From 4,
4c=-4
c=-1
g%28x%29=x%5E2-x-1
You can then find the roots of g(x) using the quadratic equation.
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+
The roots of g(x) are then
x+=+%281+%2B-+sqrt%285%29%29%2F%282%29+
or approximately, x=-0.62 and x=1.62
Second example.
A quick graph of the function shows that x=-3 and x=3 look like roots of the equation.
+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E4-2x%5E3-7x%5E2%2B18x-18%29+
h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18
h%28-3%29=81%2B54-63-54-18
h%28-3%29=0
h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18
h%283%29=81-54-63%2B54-18
h%283%29=0
So then you can rewrite your equation to look like.
h%28x%29=g%28x%29%2A%28x%2B3%29%2A%28x-3%29
where
g%28x%29=ax%5E2%2Bbx%2Bc
Follow the first example above to find a,b,and c.
Post another question if you get stuck.
Good luck.