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Polynomial functions
發問:
For each polynomial function, one zero is given. Find all others. 1) f(x)=x^3-7x^2+17x-15 ; 2-i 2) f(x)=x^4+10x^3+27x^2+10x+26 ; i
最佳解答:
(1) One zero is 2 - i, the other zero is 2 + i (x - 2 + i)(x - 2 - i) = (x - 2)^2 + 1 = x^2 - 4x + 5 Using long division, (x^3 - 7x^2 + 17x - 15)/(x^2 - 4x + 5) = (x - 3) The 3 zero's are 3, 2 + i and 2 - i (2) One zero is i, another zero is -i (x - i)(x + i) = x^2 + 1 Using long division (x^4+10x^3+27x^2+10x+26)/(x^2 + 1) = x^2 + 10x + 26 x^2 + 10x + 26 = 0 => x = [-10 +/- √(100 - 104)]/2 = -5 +/- i The 4 zero's are i, -i, -5 + i and -5 - i
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