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Sagot :
To find the polynomial function with a leading coefficient of 1 and roots [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex] each with multiplicity 1, we need to follow these steps:
1. Identify the roots:
The roots given are [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex]. Since the coefficients of a polynomial with real coefficients are real, complex roots must occur in conjugate pairs. Thus, the roots [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex] imply that the polynomial must also include the roots [tex]\( -2i \)[/tex] and [tex]\( -3i \)[/tex].
2. Construct factors corresponding to each root:
The factors corresponding to these roots are:
[tex]\[ (x - 2i), (x + 2i), (x - 3i), (x + 3i) \][/tex]
3. Form the polynomial by multiplying these factors:
[tex]\[ f(x) = (x - 2i)(x + 2i)(x - 3i)(x + 3i) \][/tex]
4. Simplify the product:
Grouping the pairs of factors:
[tex]\[ (x - 2i)(x + 2i) \quad \text{and} \quad (x - 3i)(x + 3i) \][/tex]
These are difference of squares and can be simplified as follows:
[tex]\[ (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \][/tex]
[tex]\[ (x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9 \][/tex]
Combining them, we get:
[tex]\[ f(x) = (x^2 + 4)(x^2 + 9) \][/tex]
5. Expand the product:
[tex]\[ f(x) = (x^2 + 4)(x^2 + 9) \][/tex]
Apply the distributive property (also known as FOIL for binomials):
[tex]\[ f(x) = x^2(x^2) + x^2(9) + 4(x^2) + 4(9) \][/tex]
[tex]\[ f(x) = x^4 + 9x^2 + 4x^2 + 36 \][/tex]
Combine like terms:
[tex]\[ f(x) = x^4 + 13x^2 + 36 \][/tex]
Therefore, the polynomial function with a leading coefficient of 1 and roots [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex] with multiplicity 1 is:
[tex]\[ f(x) = x^4 + 13x^2 + 36 \][/tex]
Correspondingly, the correct answer to the multiple-choice question is:
[tex]\[ f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i) \][/tex]
So, the choice:
[tex]\[ f(x) = (x+2i)(x+3i)(x-2i)(x-3i) \][/tex]
is the correct one.
1. Identify the roots:
The roots given are [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex]. Since the coefficients of a polynomial with real coefficients are real, complex roots must occur in conjugate pairs. Thus, the roots [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex] imply that the polynomial must also include the roots [tex]\( -2i \)[/tex] and [tex]\( -3i \)[/tex].
2. Construct factors corresponding to each root:
The factors corresponding to these roots are:
[tex]\[ (x - 2i), (x + 2i), (x - 3i), (x + 3i) \][/tex]
3. Form the polynomial by multiplying these factors:
[tex]\[ f(x) = (x - 2i)(x + 2i)(x - 3i)(x + 3i) \][/tex]
4. Simplify the product:
Grouping the pairs of factors:
[tex]\[ (x - 2i)(x + 2i) \quad \text{and} \quad (x - 3i)(x + 3i) \][/tex]
These are difference of squares and can be simplified as follows:
[tex]\[ (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \][/tex]
[tex]\[ (x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9 \][/tex]
Combining them, we get:
[tex]\[ f(x) = (x^2 + 4)(x^2 + 9) \][/tex]
5. Expand the product:
[tex]\[ f(x) = (x^2 + 4)(x^2 + 9) \][/tex]
Apply the distributive property (also known as FOIL for binomials):
[tex]\[ f(x) = x^2(x^2) + x^2(9) + 4(x^2) + 4(9) \][/tex]
[tex]\[ f(x) = x^4 + 9x^2 + 4x^2 + 36 \][/tex]
Combine like terms:
[tex]\[ f(x) = x^4 + 13x^2 + 36 \][/tex]
Therefore, the polynomial function with a leading coefficient of 1 and roots [tex]\( 2i \)[/tex] and [tex]\( 3i \)[/tex] with multiplicity 1 is:
[tex]\[ f(x) = x^4 + 13x^2 + 36 \][/tex]
Correspondingly, the correct answer to the multiple-choice question is:
[tex]\[ f(x) = (x + 2i)(x + 3i)(x - 2i)(x - 3i) \][/tex]
So, the choice:
[tex]\[ f(x) = (x+2i)(x+3i)(x-2i)(x-3i) \][/tex]
is the correct one.
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