[1] (x^4 - 8x^2 + 8) / (x-1) = 1 Where is the answer you want checked? What are the instructions, for this exercise ?
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[2] r=1 is a root of 6x^3 - 5x^2 + 4x - 10
False Yes, r = 1 is obviously not a root, since the symbol r does not even appear in the given polynomial.
x = 1 is not a root, either, if that's what you meant to type.
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[3] Perform the division (7x^3 -11x^2 +5x +4) / (x-1)
(7x^2 -4x + 1) + (5)/(x01) This looks correct, if x01 is supposed to mean x - 1.
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[4] Perform the division (4x^3 - 1) / (x-3)
(4x^2 + 12x +36) + -107/x-3) Is + -107 another typographical error?
You also missed a parenthesis, in the denominator. The remainder is not -107.
Question about this one... do I divde through with the 4? No. Unless you've been instructed to factor the quotient, leave it as is.
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[5] This one I'm sure is wrong, but I can't figure out how to solve it.
Find all possible rational roots x^3 + 4x^2 -3x - 5
+/- 5, 1 As you've typed it, I don't think that this polynomial has Rational roots. Did you make yet more typographical errors?
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[6] Find all roots of x^4 - x^3 -7x^2 + 5 + 10 Your answers below are incorrect, unless the polynomial term 5 is supposed to be 5x. (I'm beginning to thing that you're a sloppy typist!)
Answer: -1, 2, +/- sqrt 5
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[7] Find all roots of 6x^3 - 25x^2 -2x + 56
Answer: 2, -4/3, 7/2 These are all correct.
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[8] Find all roots of 4x^3 4x^2 -8x You did not type an operator between the first two terms.
Graphing this, the roots are -2, 0, and 1. This is correct, if the polynomial begins 4x^3 + 4x^2.
If the polynomial begins 4x^3 - 4x^2, then two of your answers are incorrect.
Using synthetic division confirms the roots 1 and -2. Substitution confirms 0. Oh, you already confirmed your result. Then I'm not sure why you posted this one.
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[9] Find all the roots of x^3 + x^2 -8x -6
Roots: -3, 2/2 + sqrt (12/2), 2/2 - sqrt (12/2)
Question - can this one be simplified? Are you kidding?! Of course 2/2 can be simplied. So can 12/2, BUT 12/2 is not the correct radicand. The only correct root that you've typed is -3.
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[10] Find the number of possible positive and negative real zeros of f(x) = x^4 - x^3 + 2x^2 + x -5
Sign changes with f(x) = 2 or 0 This statement makes no sense; there can be only one number of sign changes.
BTW, this exercise does not ask for the number of sign changes. It's asking about the number of possible Real roots.
The number of sign changes in f(x) is neither 2 nor 0. There are three sign changes, period. So, what does that tell you about the number of positive Real zeros?
Sign changes with f(-x) = 2 or 0Again, f(-x) does not have 2 or 0 sign changes. It has one sign change.
Please review Descartes' Rule of Signs, and try again.
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[11] Find the number of possible postive and negative real zeros of f(x)=3x^3 - 4x^2 - 5x + 1
Number of sign changes when f(x) = 2 or 0 Now you're including the word "when". These statements make even less sense than above.
We don't want to consider when the function equals 2, when discussing roots!
Number of sign changes when f(-x) 1 Are you trying to tell us that f(-x) has one sign change? Please be clear.
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