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Sagot :
To find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros, we proceed as follows:
1. Identify the given zeros and their conjugates:
- The given zeros are [tex]\(3 - i\)[/tex] and [tex]\(\sqrt{6}\)[/tex].
- Since the polynomial must have rational coefficients, the complex zero [tex]\(3 - i\)[/tex] must be paired with its conjugate [tex]\(3 + i\)[/tex].
- Similarly, the irrational zero [tex]\(\sqrt{6}\)[/tex] must be paired with its conjugate [tex]\(-\sqrt{6}\)[/tex].
2. Form factors of the polynomial:
- For each pair of zeros, we form a factor of the polynomial. The factors corresponding to the given zeros and their conjugates are:
[tex]\[ (x - (3 - i))(x - (3 + i)) \][/tex]
[tex]\[ (x - \sqrt{6})(x + \sqrt{6}) \][/tex]
3. Expand the factors:
- First, we expand the factor for the complex conjugate zeros:
[tex]\[ (x - (3 - i))(x - (3 + i)) = (x - 3 + i)(x - 3 - i) \][/tex]
[tex]\[ = ((x - 3) + i)((x - 3) - i) \][/tex]
[tex]\[ = (x - 3)^2 - i^2 \][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[ = (x - 3)^2 - (-1) \][/tex]
[tex]\[ = (x - 3)^2 + 1 \][/tex]
[tex]\[ = x^2 - 6x + 9 + 1 \][/tex]
[tex]\[ = x^2 - 6x + 10 \][/tex]
- Next, we expand the factor for the irrational conjugate zeros:
[tex]\[ (x - \sqrt{6})(x + \sqrt{6}) \][/tex]
[tex]\[ = x^2 - (\sqrt{6})^2 \][/tex]
[tex]\[ = x^2 - 6 \][/tex]
4. Form the polynomial by multiplying the factors:
- Now, to get the polynomial function, we need to multiply the expanded factors together:
[tex]\[ f(x) = (x^2 - 6x + 10)(x^2 - 6) \][/tex]
5. Expand the resulting polynomial:
[tex]\[ f(x) = (x^2 - 6x + 10)(x^2 - 6) \][/tex]
We distribute each term of the first polynomial to each term of the second polynomial:
[tex]\[ = x^2(x^2 - 6) - 6x(x^2 - 6) + 10(x^2 - 6) \][/tex]
[tex]\[ = (x^4 - 6x^2) - (6x^3 - 36x) + (10x^2 - 60) \][/tex]
[tex]\[ = x^4 - 6x^2 - 6x^3 + 36x + 10x^2 - 60 \][/tex]
6. Combine like terms:
[tex]\[ = x^4 - 6x^3 + 4x^2 + 36x - 60 \][/tex]
Therefore, the polynomial function in expanded form is:
[tex]\[ f(x) = x^4 - 6x^3 + 4x^2 + 36x - 60. \][/tex]
1. Identify the given zeros and their conjugates:
- The given zeros are [tex]\(3 - i\)[/tex] and [tex]\(\sqrt{6}\)[/tex].
- Since the polynomial must have rational coefficients, the complex zero [tex]\(3 - i\)[/tex] must be paired with its conjugate [tex]\(3 + i\)[/tex].
- Similarly, the irrational zero [tex]\(\sqrt{6}\)[/tex] must be paired with its conjugate [tex]\(-\sqrt{6}\)[/tex].
2. Form factors of the polynomial:
- For each pair of zeros, we form a factor of the polynomial. The factors corresponding to the given zeros and their conjugates are:
[tex]\[ (x - (3 - i))(x - (3 + i)) \][/tex]
[tex]\[ (x - \sqrt{6})(x + \sqrt{6}) \][/tex]
3. Expand the factors:
- First, we expand the factor for the complex conjugate zeros:
[tex]\[ (x - (3 - i))(x - (3 + i)) = (x - 3 + i)(x - 3 - i) \][/tex]
[tex]\[ = ((x - 3) + i)((x - 3) - i) \][/tex]
[tex]\[ = (x - 3)^2 - i^2 \][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[ = (x - 3)^2 - (-1) \][/tex]
[tex]\[ = (x - 3)^2 + 1 \][/tex]
[tex]\[ = x^2 - 6x + 9 + 1 \][/tex]
[tex]\[ = x^2 - 6x + 10 \][/tex]
- Next, we expand the factor for the irrational conjugate zeros:
[tex]\[ (x - \sqrt{6})(x + \sqrt{6}) \][/tex]
[tex]\[ = x^2 - (\sqrt{6})^2 \][/tex]
[tex]\[ = x^2 - 6 \][/tex]
4. Form the polynomial by multiplying the factors:
- Now, to get the polynomial function, we need to multiply the expanded factors together:
[tex]\[ f(x) = (x^2 - 6x + 10)(x^2 - 6) \][/tex]
5. Expand the resulting polynomial:
[tex]\[ f(x) = (x^2 - 6x + 10)(x^2 - 6) \][/tex]
We distribute each term of the first polynomial to each term of the second polynomial:
[tex]\[ = x^2(x^2 - 6) - 6x(x^2 - 6) + 10(x^2 - 6) \][/tex]
[tex]\[ = (x^4 - 6x^2) - (6x^3 - 36x) + (10x^2 - 60) \][/tex]
[tex]\[ = x^4 - 6x^2 - 6x^3 + 36x + 10x^2 - 60 \][/tex]
6. Combine like terms:
[tex]\[ = x^4 - 6x^3 + 4x^2 + 36x - 60 \][/tex]
Therefore, the polynomial function in expanded form is:
[tex]\[ f(x) = x^4 - 6x^3 + 4x^2 + 36x - 60. \][/tex]
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