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Sagot :
To factor the polynomial [tex]\( f(x) = x^4 - 4x^3 - 8x^2 - 16x - 48 \)[/tex] completely, given that [tex]\(-2i\)[/tex] is a zero, we can use the Conjugate Root Theorem along with polynomial division. Here is a step-by-step solution:
1. Identify Given and Conjugate Roots:
If [tex]\(-2i\)[/tex] is a zero, then by the Conjugate Root Theorem, [tex]\(2i\)[/tex] (the conjugate) is also a zero.
2. Form a Quadratic Factor:
The zeros [tex]\(-2i\)[/tex] and [tex]\(2i\)[/tex] can be used to form a quadratic factor. These roots give us the factor:
[tex]\[ (x - (-2i))(x - 2i) = (x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \][/tex]
Therefore, [tex]\(x^2 + 4\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
3. Perform Polynomial Division:
Next, we need to divide [tex]\(f(x)\)[/tex] by [tex]\(x^2 + 4\)[/tex] to find the other factor. We'll use polynomial long division for this.
- Divide the leading term of [tex]\(x^4 - 4x^3 - 8x^2 - 16x - 48\)[/tex] by the leading term of [tex]\(x^2 + 4\)[/tex]:
[tex]\[ \frac{x^4}{x^2} = x^2 \][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (x^4 - 4x^3 - 8x^2 - 16x - 48) - (x^4 + 4x^2) = -4x^3 - 12x^2 - 16x - 48 \][/tex]
- Repeat the division process:
[tex]\[ \frac{-4x^3}{x^2} = -4x \][/tex]
- Multiply [tex]\(-4x\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (-4x^3 - 12x^2 - 16x - 48) - (-4x^3 - 16x) = -12x^2 \][/tex]
- Repeat the division process again:
[tex]\[ \frac{-12x^2}{x^2} = -12 \][/tex]
- Multiply [tex]\(-12\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (-12x^2 - 48) - (-12x^2 - 48) = 0 \][/tex]
The quotient from the division is [tex]\(x^2 - 4x - 12\)[/tex], and the remainder is 0.
Therefore, we have:
[tex]\[ f(x) = (x^2 + 4)(x^2 - 4x - 12) \][/tex]
4. Factor the Quadratic Quotient:
Now we need to factor the quadratic [tex]\(x^2 - 4x - 12\)[/tex]. To do this, find two numbers that multiply to [tex]\(-12\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(2\)[/tex].
Therefore, we can factor [tex]\(x^2 - 4x - 12\)[/tex] as:
[tex]\[ x^2 - 4x - 12 = (x - 6)(x + 2) \][/tex]
5. Combine All Factors:
Putting it all together, the complete factorization of the polynomial is:
[tex]\[ f(x) = (x^2 + 4)(x - 6)(x + 2) \][/tex]
Thus, the polynomial function [tex]\( f(x) = x^4 - 4x^3 - 8x^2 - 16x - 48 \)[/tex] factors completely as:
[tex]\[ f(x) = (x^2 + 4)(x - 6)(x + 2) \][/tex]
1. Identify Given and Conjugate Roots:
If [tex]\(-2i\)[/tex] is a zero, then by the Conjugate Root Theorem, [tex]\(2i\)[/tex] (the conjugate) is also a zero.
2. Form a Quadratic Factor:
The zeros [tex]\(-2i\)[/tex] and [tex]\(2i\)[/tex] can be used to form a quadratic factor. These roots give us the factor:
[tex]\[ (x - (-2i))(x - 2i) = (x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \][/tex]
Therefore, [tex]\(x^2 + 4\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
3. Perform Polynomial Division:
Next, we need to divide [tex]\(f(x)\)[/tex] by [tex]\(x^2 + 4\)[/tex] to find the other factor. We'll use polynomial long division for this.
- Divide the leading term of [tex]\(x^4 - 4x^3 - 8x^2 - 16x - 48\)[/tex] by the leading term of [tex]\(x^2 + 4\)[/tex]:
[tex]\[ \frac{x^4}{x^2} = x^2 \][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (x^4 - 4x^3 - 8x^2 - 16x - 48) - (x^4 + 4x^2) = -4x^3 - 12x^2 - 16x - 48 \][/tex]
- Repeat the division process:
[tex]\[ \frac{-4x^3}{x^2} = -4x \][/tex]
- Multiply [tex]\(-4x\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (-4x^3 - 12x^2 - 16x - 48) - (-4x^3 - 16x) = -12x^2 \][/tex]
- Repeat the division process again:
[tex]\[ \frac{-12x^2}{x^2} = -12 \][/tex]
- Multiply [tex]\(-12\)[/tex] by [tex]\(x^2 + 4\)[/tex] and subtract:
[tex]\[ (-12x^2 - 48) - (-12x^2 - 48) = 0 \][/tex]
The quotient from the division is [tex]\(x^2 - 4x - 12\)[/tex], and the remainder is 0.
Therefore, we have:
[tex]\[ f(x) = (x^2 + 4)(x^2 - 4x - 12) \][/tex]
4. Factor the Quadratic Quotient:
Now we need to factor the quadratic [tex]\(x^2 - 4x - 12\)[/tex]. To do this, find two numbers that multiply to [tex]\(-12\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(2\)[/tex].
Therefore, we can factor [tex]\(x^2 - 4x - 12\)[/tex] as:
[tex]\[ x^2 - 4x - 12 = (x - 6)(x + 2) \][/tex]
5. Combine All Factors:
Putting it all together, the complete factorization of the polynomial is:
[tex]\[ f(x) = (x^2 + 4)(x - 6)(x + 2) \][/tex]
Thus, the polynomial function [tex]\( f(x) = x^4 - 4x^3 - 8x^2 - 16x - 48 \)[/tex] factors completely as:
[tex]\[ f(x) = (x^2 + 4)(x - 6)(x + 2) \][/tex]
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