If -1 and -3 are two zeros of the polynomial x4+5x3+8x2+7x+3 then find the other zeros if they exist
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Given polynomial,
p(x)=x^4+5x^3+8x^2+7x+3
Given that , -1 and -3 are zeroes of p(x)
Then (x+1) and (x+3) are factors of p(x)
(x+1)(x+3)=x^2+3x+x+3=x^2+4x+3
Then,(x^2+4x+3) is also a factor of p(x).
Now,
(x^4+5x^3+8x^2+7x+3)/(x^2+4x+3)
=x^2+x+1
Let us find out discriminant of above equation,
Discriminant=b^2-4ac
=(1)^2-4(1)(1)
=1-4
=-3
=-3<0
Here discriminant is less than zero.
Therefore equation (x^2+x+1) has no real roots , it has imaginary roots.
Hence zeroes of given polynomial are -1 and -3 , it has no other real roots , other two roots are imagianary.
p(x)=x^4+5x^3+8x^2+7x+3
Given that , -1 and -3 are zeroes of p(x)
Then (x+1) and (x+3) are factors of p(x)
(x+1)(x+3)=x^2+3x+x+3=x^2+4x+3
Then,(x^2+4x+3) is also a factor of p(x).
Now,
(x^4+5x^3+8x^2+7x+3)/(x^2+4x+3)
=x^2+x+1
Let us find out discriminant of above equation,
Discriminant=b^2-4ac
=(1)^2-4(1)(1)
=1-4
=-3
=-3<0
Here discriminant is less than zero.
Therefore equation (x^2+x+1) has no real roots , it has imaginary roots.
Hence zeroes of given polynomial are -1 and -3 , it has no other real roots , other two roots are imagianary.
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