Given that 2+i is a root of the polynomial f(x) = x^3 - 3x^2 + x + 5, we can find the other roots by using the complex conjugate theorem.
Since f(x) has real coefficients, any complex roots must come in conjugate pairs. This means that if 2+i is a root, then the conjugate 2-i must also be a root.
To find the third root, we can use the fact that the sum of the roots of a polynomial is equal to the negative of the coefficient of the second-to-last term divided by the leading coefficient. In this case, the sum of the roots is -(-3)/1 = 3.
Since we know 2+i and 2-i are two of the roots, we can find the third root by subtracting the sum of those two roots from the total sum of the roots:
3 - (2+i) - (2-i) = 3 - 2 - i - 2 + i = -1
Therefore, the other roots of the polynomial f(x) are 2+i, 2-i, and -1.
4, if a root of f(x)=x^3-3x^2+x+5 is 2+i. find the other roots of f.
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