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Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and roots of 2-√6, 2+√6 and 5-i

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Final answer:

To find the polynomial function of the least degree with real coefficients and the given roots 2-√6, 2+√6, and 5-i, we must first include the complex conjugate root of 5+i. By factoring and expanding these roots, we construct the polynomial function x^4 - 14x^3 + 52x^2 - 20x - 12.

Step-by-step explanation:

The subject of this question is to find a polynomial function of the least degree with real coefficients, a leading coefficient of 1, and the given roots 2-√6, 2+√6, and 5-i. We know that complex roots must come in conjugate pairs for the polynomial to have real coefficients, so the missing conjugate root is 5+i. Now we can construct the factors of the polynomial from its roots: (x - (2-√6))(x - (2+√6))(x - (5-i))(x - (5+i)).

Expanding these factors, we have:

  • (x - 2 + √6)(x - 2 - √6) = (x - 2)2 - (√6)2
  • (x - 5 + i)(x - 5 - i) = (x - 5)2 + 1

Multiplying these together, we obtain the polynomial:

(x2 - 4x + 4 - 6)(x2 - 10x + 26) = x4 - 14x3 + 52x2 - 20x - 12

This is the polynomial function of the least degree that satisfies the given conditions.

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