SOLUTION: For the following polynomials , find all real and complex roots .(hint: Use the rational root theorem to identify potential roots. 1- y(x) = x^3-1 2- h(x) = x^4 -1 3- g(x) = x

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Question 936620: For the following polynomials , find all real and complex roots .(hint: Use the rational root theorem to identify potential roots.
1- y(x) = x^3-1
2- h(x) = x^4 -1
3- g(x) = x^4 +2x^3-16x^2-2x+15
4- f(x) = x^4+ 2x^3 -16x^2 -2x +15
(note:what they mean by potential roots?) .Thanks
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
to identify POTENTIAL roots use THE RATIONAL ZERO THEOREM
If f%28x%29 =a%5Bn%5Dx%5En+ .... .+a%5B1%5Dx+a%5B0%5D has integer coefficients, then every rational zero of f%28x%29 has the following form:
p%2Fq=%28factors_of_constant_term_a%5B0%5D%29%2F%28factors_of_leading_coefficient_a%29
using that, we have:
1- y%28x%29+=+x%5E3-1
--> factor 1 and 1:
x+= ±1%2F1 which is x = ±1%2F1
2- h%28x%29+=+x%5E4+-1
--> factor 1 and 1:
x+= ±1%2F1 which is x = ±1%2F1
3- g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15
--> factor 1, and 15:
1 X 1,3,5,15
x = ±1%2F1, ±3%2F1,±5%2F1,±15%2F1, which is => x = ±1,±3,±5,±15
Test these zeros using synthetic division.
test x=1
g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15

1|--1-----2----..-16------...-2----15
---| ------1----..-1----..-1------17---..-15
---| 1---- 1----.. -17--------15------- 0
so, x=1 is the root
same way you check all other potential roots

now find roots:

1- y%28x%29+=+x%5E3-1........factor: set y%28x%29=0 and use a%5E3+-+b%5E3+=+%28a+-b%29%28a%5E2+%2B+ab+%2B+b%5E2%29++..(difference of 2 cubes rule)
0=+%28x-1%29%28x%5E2%2Bx%2B1%29
we already know one root: if 0=+%28x-1%29=> x=1
use quadratic formula to find other two roots from 0=+%28x%5E2%2Bx%2B1%29:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+1-4+%29%29%2F2+
x+=+%28-1+%2B-+sqrt%28+-3+%29%29%2F2+
solution will be complex roots:
x+=+-1%2F2+%2B+sqrt%28+-3+%29%2F2+
and
x+=+-1%2F2+-+sqrt%28+-3+%29%2F2+

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E3-1%29+

2- h%28x%29+=+x%5E4+-1
0+=+x%5E4+-1%5E4
0+=+%28x%5E2%29%5E2+-%281%5E2%29%5E2
0+=+%28x%5E2+-1%5E2%29%28x%5E2+%2B1%5E2%29
%28x-1%29%28x%2B1%29%28x%5E2%2B1%29=0+
roots:


%28x-1%29=0+ =>x=1
%28x%2B1%29=0+=>x=-1......real roots

%28x%5E2%2B1%29=0+=> x%5E2=-1+=> x=sqrt%28-1%29+=> x=i+ or x=-i+
x=1.........complex roots
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%28x-1%29%28x%2B1%29%28x%5E2%2B1%29%29+

3- g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15...write 2x%5E3 as 5x%5E3-3x%5E3 and -16x%5E2 as -15x%5E2-x%5E2
0+=+x%5E4-x%5E2%2B5x%5E3-5x-3x%5E3-3x-15x%5E2-15....group
0+=+%28x%5E4-x%5E2%29%2B%285x%5E3-5x%29-%283x%5E3-3x%29-%2815x%5E2-15%29
0+=+x%5E2%28x%5E2-1%29%2B5x%28x%5E2-1%29-3x%28x%5E2-1%29-15%28x%5E2-1%29
0+=+%28x%5E2-1%29%28x%5E2%2B5x-3x-15%29...group
0+=+%28x-1%29%28x%2B1%29%28%28x%5E2-3x%29%2B%285x-15%29%29
0+=+%28x-1%29%28x%2B1%29%28x%28x-3%29%2B5%28x-3%29%29
0+=+%28x-1%29%28x%2B1%29%28x-3%29%28x%2B5%29
roots:
0+=+%28x-1%29=> x=1
0+=%28x%2B1%29=> x=-1
0+=+%28x-3%29=>x=3
0+=+%28x%2B5%29=> x=-5........all roots are real roots

+graph%28+600%2C+600%2C+-25%2C+25%2C+-45%2C+25%2C+x%5E4%2B+2x%5E3+-16x%5E2+-2x+%2B15%29+


4- f%28x%29+=+x%5E4%2B+2x%5E3+-16x%5E2+-2x+%2B15+ this is same as 3-