Question: Choose a polynomial function in standard form with real coefficients whose zeros are 2 and - 6 and multiplicity of 3 and 2 respectively.
A ) f(x) = x5+80x2
B ) f(x) = - 24x3+80x2
C ) f(x) = x5+6x4+336x-288
D ) f(x) = x5+6x4-24x3-80x2+336x-288
Steps to derive
1 The factor corresponding to the zero 2 with multiplicity 3 is (x - 2)3.
2 The factor corresponding to the zero - 6 with multiplicity 2 is (x + 6)2.
3 Hence, the polynomial function with given zeros is f (x) = (x - 2)3(x + 6)2
4 = (x3-6x2+12x - 8)(x2+12x + 36)
5 = x5+6x4-24x3-80x2+336x-288
[Simplify.]
6 So, the polynomial function with real coefficients having the given zeros in the standard form is f(x) = x5+6x4-24x3-80x2+336x-288.
Hence the right answer is Option D
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