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Sagot :
Let's find the polynomial function of degree 4 with only real coefficients, given the roots [tex]\(-2.4\)[/tex] and [tex]\(2i\)[/tex] and the condition [tex]\(f(-3) = 15\)[/tex].
### Step-by-step Solution:
1. Identify the Roots:
- The given roots are [tex]\(-2.4\)[/tex] and [tex]\(2i\)[/tex].
- Since the polynomial must have real coefficients, the complex root [tex]\(2i\)[/tex] implies the presence of its complex conjugate [tex]\(-2i\)[/tex].
Thus, the roots are [tex]\(-2.4\)[/tex], [tex]\(2.4\)[/tex], [tex]\(2i\)[/tex], and [tex]\(-2i\)[/tex].
2. Form the Polynomial:
- We start by writing the polynomial in its factored form:
[tex]\[ P(x) = a(x + 2.4)(x - 2.4)(x - 2i)(x + 2i) \][/tex]
- Note that [tex]\((x - 2i)(x + 2i) = x^2 + 4\)[/tex] and [tex]\((x + 2.4)(x - 2.4) = x^2 - 5.76\)[/tex].
Thus, the polynomial can be rewritten as:
[tex]\[ P(x) = a(x^2 - 5.76)(x^2 + 4) \][/tex]
3. Expand the Polynomial:
- Multiply the two quadratic expressions:
[tex]\[ P(x) = a[(x^2 - 5.76)(x^2 + 4)] \][/tex]
[tex]\[ P(x) = a(x^4 + 4x^2 - 5.76x^2 - 23.04) \][/tex]
[tex]\[ P(x) = a(x^4 - 1.76x^2 - 23.04) \][/tex]
4. Find the Leading Coefficient [tex]\(a\)[/tex]:
- We are given that [tex]\(P(-3) = 15\)[/tex].
Substituting [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[ P(-3) = a((-3)^4 - 1.76(-3)^2 - 23.04) = 15 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ (-3)^4 = 81 \][/tex]
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ P(-3) = a(81 - 1.76 \cdot 9 - 23.04) \][/tex]
Simplify the middle term:
[tex]\[ 1.76 \cdot 9 = 15.84 \][/tex]
Therefore:
[tex]\[ P(-3) = a(81 - 15.84 - 23.04) \][/tex]
[tex]\[ P(-3) = a(81 - 38.88) \][/tex]
[tex]\[ P(-3) = a(42.12) = 15 \][/tex]
- Solve for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{15}{42.12} \][/tex]
[tex]\[ a = 0.356125356125356 \][/tex]
5. Form the Final Polynomial:
- Substitute [tex]\(a\)[/tex] back into the polynomial:
[tex]\[ P(x) = 0.356125356125356(x^4 - 1.76x^2 - 23.04) \][/tex]
So, the final polynomial in its standard form is:
[tex]\[ P(x) = 0.356125356125356x^4 - 0.626780626780627x^2 - 8.2051282051282 \][/tex]
This is the polynomial function of degree 4 with the given properties.
### Step-by-step Solution:
1. Identify the Roots:
- The given roots are [tex]\(-2.4\)[/tex] and [tex]\(2i\)[/tex].
- Since the polynomial must have real coefficients, the complex root [tex]\(2i\)[/tex] implies the presence of its complex conjugate [tex]\(-2i\)[/tex].
Thus, the roots are [tex]\(-2.4\)[/tex], [tex]\(2.4\)[/tex], [tex]\(2i\)[/tex], and [tex]\(-2i\)[/tex].
2. Form the Polynomial:
- We start by writing the polynomial in its factored form:
[tex]\[ P(x) = a(x + 2.4)(x - 2.4)(x - 2i)(x + 2i) \][/tex]
- Note that [tex]\((x - 2i)(x + 2i) = x^2 + 4\)[/tex] and [tex]\((x + 2.4)(x - 2.4) = x^2 - 5.76\)[/tex].
Thus, the polynomial can be rewritten as:
[tex]\[ P(x) = a(x^2 - 5.76)(x^2 + 4) \][/tex]
3. Expand the Polynomial:
- Multiply the two quadratic expressions:
[tex]\[ P(x) = a[(x^2 - 5.76)(x^2 + 4)] \][/tex]
[tex]\[ P(x) = a(x^4 + 4x^2 - 5.76x^2 - 23.04) \][/tex]
[tex]\[ P(x) = a(x^4 - 1.76x^2 - 23.04) \][/tex]
4. Find the Leading Coefficient [tex]\(a\)[/tex]:
- We are given that [tex]\(P(-3) = 15\)[/tex].
Substituting [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[ P(-3) = a((-3)^4 - 1.76(-3)^2 - 23.04) = 15 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ (-3)^4 = 81 \][/tex]
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ P(-3) = a(81 - 1.76 \cdot 9 - 23.04) \][/tex]
Simplify the middle term:
[tex]\[ 1.76 \cdot 9 = 15.84 \][/tex]
Therefore:
[tex]\[ P(-3) = a(81 - 15.84 - 23.04) \][/tex]
[tex]\[ P(-3) = a(81 - 38.88) \][/tex]
[tex]\[ P(-3) = a(42.12) = 15 \][/tex]
- Solve for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{15}{42.12} \][/tex]
[tex]\[ a = 0.356125356125356 \][/tex]
5. Form the Final Polynomial:
- Substitute [tex]\(a\)[/tex] back into the polynomial:
[tex]\[ P(x) = 0.356125356125356(x^4 - 1.76x^2 - 23.04) \][/tex]
So, the final polynomial in its standard form is:
[tex]\[ P(x) = 0.356125356125356x^4 - 0.626780626780627x^2 - 8.2051282051282 \][/tex]
This is the polynomial function of degree 4 with the given properties.
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