Please help me with these 2 questions! They are the only ones I don't know. Thanks in advance!!
1) Find the equation of a function that has the following zeros where f(0)=52; The zeros are -2, 8, 7 and -4i
2) Use the Remainder Theorem to find the remainder when f(x) is divided by x - k. Is x - k a factor of f(x)? The equation is x + 3; 4x^3 + 6x^2 - 11x + 21
1) Find the equation of a function that has the following zeros where f(0)=52; The zeros are -2, 8, 7 and -4i
2) Use the Remainder Theorem to find the remainder when f(x) is divided by x - k. Is x - k a factor of f(x)? The equation is x + 3; 4x^3 + 6x^2 - 11x + 21
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1) If f(0) is going to equal 52, then the coefficients of the function are likely to be real, in which case, +4i is also a zero. Use the factor theorem to convert the five zeros into factors:
(x + 2) (x - 8) (x - 7) (x + 4i) (x - 4i)
Multiply the factors together. (It's easier to multiply the complex factors first.):
x^5 - 13x^4 + 42x^3 - 96x^2 + 416x + 1792
f(0) for this function is 1792, so multiply the whole function by (52/1792) = (13/448) to achieve the desired f(0):
f(x) = (13/448) (x^5 - 13x^4 + 42x^3 - 96x^2 + 416x + 1792)
2) The remainder theorem tells you that the remainder after dividing f(x) by x - k will be f(k). So in the present case, evaluate f(-3):
4(-3)^3 + 6(-3)^2 - 11(-3) + 21 = -108 + 54 + 33 + 21 = 0
Since the remainder is zero, x+3 is indeed a factor of 4x^3 + 6x^2 - 11x + 21.
(x + 2) (x - 8) (x - 7) (x + 4i) (x - 4i)
Multiply the factors together. (It's easier to multiply the complex factors first.):
x^5 - 13x^4 + 42x^3 - 96x^2 + 416x + 1792
f(0) for this function is 1792, so multiply the whole function by (52/1792) = (13/448) to achieve the desired f(0):
f(x) = (13/448) (x^5 - 13x^4 + 42x^3 - 96x^2 + 416x + 1792)
2) The remainder theorem tells you that the remainder after dividing f(x) by x - k will be f(k). So in the present case, evaluate f(-3):
4(-3)^3 + 6(-3)^2 - 11(-3) + 21 = -108 + 54 + 33 + 21 = 0
Since the remainder is zero, x+3 is indeed a factor of 4x^3 + 6x^2 - 11x + 21.
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If you don't include (13/448) in the answer to #1, you don't get f(0)=52. If you're asking whether to multiply through to get f(x) = 13x^5/448 - 169x^4/448 ... , that's up to you, it's the same either way.
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so the final answer is: f(x)=(13/448)(x^5 - 13x^4 + 42x^3 - 96x^2 + 416x + 1792)?
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