Solve 2x^5+x4-12x^3-12x3-12x^2+x+2
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ANSWER
2x
5
+x
4
−12x
3
−12x
2
+x+2=0
If we look at the coefficients, they are 2,1,−12,−12,1,2
Since they are the same for x
5
and x
0
,x
1
and x
4
,x
2
and x
3
, we conclude that x+1 is a factor of the polynomial.
∴(x+1)(2x
4
−x
3
−11x
2
−x+2)=0
Now, we look at 2x
4
−x
3
−11x
2
−x+2=0.
Dividing throughout by x
2
, we have 2x
2
+
x
2
2
−x−
x
1
−11=0
Substituting x+
x
1
=a, we have
2(a
2
−2)−a−11=0
∴2a
2
−a−15=0
∴2a
2
−6a+5a−15=0
∴(2a+5)(a−3)=0
Resubstituting a=x+
x
1
and multiplying by x
2
, we have
(2x
2
+5x+2)(x
2
−3x+1)=0
⇒2x
5
+x
4
−12x
3
−12x
2
+x+2=(x+1)(2x
2
+5x+2)(x
2
−3x+1)=0
∴(x+1)(2x+1)(x+2)(x
2
−3x+1)=0
∴x=−1,−
2
1
,−2,
2
3+
5
,
2
3−
5
Step-by-step explanation:
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