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The polynomial f(x)=x^4+a x^3+b x^2+c x+...

The polynomial `f(x)=x^4+a x^3+b x^2+c x+d` has real coefficients and `f(2i)=f(2+i)=0.` Find the value of `(a+b+c+d)dot`

Text Solution

Verified by Experts

The correct Answer is:
9
If a polynomial has real coefficients, then roots occur in complex
conjugate and roots are `pm 2i, 2 i.` Hence.
` f (x) = (x + 2i) (x - 2i) (x -2-i)(x -2-i)`
`therefore f(1) = (1 + 2i) (1 + 2i) (1 - 2-i)(1 - 2 + i)`
` 5xx2 = 10 `
Also, ` f(1) = 1 + a + b + c + d`
`therefore 1 + a + b + c + d = 10`
`rArr a + b + c + d = 9` .
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