Factor f(x) into linear factors given that k is a zero of f(x).
f(x)=x^3+(12-4i)x^2+(32-48i)x-128i, k=4i
In completely factored form, f(x) =____
(factor completely. simplify your answer)
so, if you divide f(x) by (x-4i) you get
f(x) = (x-4i)(x^2+12x+32) = (x-4i)(x+4)(x+8)
Factor f(x) into linear factors given that k is a zero of f(x).
f(x)=3x3+ 14x2−77x+24; k=3
f(x)=enter your response here (Factor completely.)
To factor f(x) into linear factors, given that k=4i is a zero of f(x), we can use the factor theorem.
According to the factor theorem, if k is a root of f(x), then (x - k) is a factor of f(x). So, in this case, (x - 4i) is a factor of f(x).
To find the other factor, we can use polynomial long division or synthetic division. Since f(x) is a cubic polynomial, we will use polynomial long division.
Performing polynomial long division, dividing f(x) by (x - 4i), the process is as follows:
x^2 + (8 + 2i)x + 16i
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x - 4i | x^3 + (12 - 4i)x^2 + (32 - 48i)x - 128i
- x^3 + 4ix^2
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(12 - 4i)x^2 + (32 - 48i)x
- (12 - 4i)x^2 + 48ix
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(32 - 48i)x - 48ix
- (32 - 48i)x + 128i
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- 128i + 128i
Since the remainder is zero, it means that (x - 4i) is a factor of f(x).
Now, we can write f(x) as a product of its factors:
f(x) = (x - 4i)(x^2 + (8 + 2i)x + 16i)
To factor the quadratic factor, we can use the quadratic formula or complete the square. Let's use the quadratic formula.
The quadratic factor can be written as:
x^2 + (8 + 2i)x + 16i = 0
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
a = 1, b = (8 + 2i), c = 16i
x = (-(8 + 2i) ± √((8 + 2i)^2 - 4(1)(16i)))/(2(1))
Simplifying:
x = (-8 - 2i ± √(64 + 16i - 16 + 64i - 64))/(2)
x = (-8 - 2i ± √(64i))/(2)
x = (-8 - 2i ± 8i)/(2)
x = (-8 - 2i ± 8i)/(2)
x = (-8 ± 6i)/(2)
x = -4 ± 3i
Therefore, the completely factored form of f(x) is:
f(x) = (x - 4i)(x + 4 - 3i)(x + 4 + 3i)
To factor the polynomial f(x) = x^3 + (12 - 4i)x^2 + (32 - 48i)x - 128i when k = 4i is a zero, we can use synthetic division.
1. First, set up the synthetic division as follows:
4i | 1 12 - 4i 32 - 48i -128i
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2. Divide the leading coefficient, 1, by 4i to get 1/4i, which simplifies to -i/4.
-i/4 | 1 12 - 4i 32 - 48i -128i
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3. Multiply -i/4 by 4i to get -1, and add it to 12 to get 11:
-i/4 | 1 12 - 4i 32 - 48i -128i
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4. Multiply -i/4 by 11 to get -11i/4, and add it to 32 - 48i to get 32 - 48i - 11i/4.
Simplify this expression:
-i/4 | 1 12 - 4i 32 - 48i -128i
| -11i/4
5. Multiply -i/4 by 32 - 48i - 11i/4 to get (32i - 48) - (11/4)i, and add it to -128i to get -128i + (32i - 48) - (11/4)i.
Simplify this expression:
-i/4 | 1 12 - 4i 32 - 48i -128i
| -11i/4 32i - 48 -11/4i
6. Perform the final multiplication and addition to obtain:
-i/4 | 1 12 - 4i 32 - 48i -128i
| -11i/4 32i - 48 -11/4i
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1 -11/4i 32i - 48 -11/4i - 128i
Therefore, the factored form of f(x) is:
f(x) = (x - 4i)(x^2 - (11/4i)x + 32i - 48 - 11/4i - 128i)
Simplifying further, we can combine like terms:
f(x) = (x - 4i)(x^2 - (11/4i)x - 192 - (15/4)i)