course ??????~????????assignment #029029. Query 29
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15:16:36 4.2.20 (was 4.2.10). If f(x)= (x^2-5) / x^3 a polynomial?
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RESPONSE --> f(x)= (x^2-5) / x^3 = x^2/x^3 - 5/x^3 yes it is a polynomial function confidence assessment: 3
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15:17:29 This is not a polynomial function. It is the ratio of two polynomials, the ratio of x^2 - 5 to x^3.
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RESPONSE --> Ok self critique assessment: 3
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15:25:37 4.2.40 (was 4.2.30). If a polynomial has zeros at x = -4, 0, 2 the what is its minimum possible degree, and what form will the polynomial have?
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RESPONSE --> f(x) = a(x -(-4))(x-0)(x-2) for a = 1, f(x) = (x+4)(x)(x-2) x(x+4)(x-2) = x(x^2 + 2x - 8) = x^3 + 2x^2 - 8x minimum possible degree is 3 confidence assessment: 2
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15:25:55 The factors of the polynomial must include (x - (-4) ) = x + 4, x - 0 = x, and x - 2. So the polynomial must be a multiply of (x+4)(x)(x-2). The general form of the polynomial is therefore f(x)=a(x+4)(x-0)(x-2).
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RESPONSE --> ok self critique assessment: 3
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15:28:02 4/2/52 (was 4.2.40). What are the zeros of the function f(x)=(x+sqrt(3))^2 (x-2)^4 and what is the multiplicity of each?
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RESPONSE --> The zeros are -sqrt(3) with the multiplicity of 2 and 2 with the multiplicity of 4 confidence assessment: 2
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15:28:18 f(x) will be zero if x + sqrt(3) = 0 or if x - 2 = 0. The solutions to these equations are x = - sqrt(3) and x = 2. The zero at x = -sqrt(3) comes from (x + sqrt(3))^2 so has degree 2. The zero at x = 2 comes from (x-2)^4 so has degree 4.
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RESPONSE --> ok self critique assessment: 3
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15:29:26 For each zero does the graph touch or cross the x axis?
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RESPONSE --> For -sqrt(3) the graph touches the x axis, and for 2 the graph crosses the x-axis confidence assessment: 3
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15:30:19 In each case the zero is of even degree, so it just touches the x axis. Near x = -sqrt(3) the graph is nearly a constant multiple of (x+sqrt(3))^2. Near x = 2 the graph is nearly a constant multiple of (x - 2)^4.
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RESPONSE --> I'm sorry, I got confused. Yes, the graph touches the x axis in both cases since both degrees are even. self critique assessment: 3
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15:31:29 What power function does the graph of f resemble for large values of | x | ?
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RESPONSE --> The function resembles y = x^6 for large values of |x| confidence assessment: 3
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15:31:42 If you multiply out all the terms you will be a polynomial with x^6 as the highest-power term, i.e., the 'leading term'. For large | x | the polynomial resembles this 'leading term'--i.e., it resembles the power function x^6. **
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RESPONSE --> ok self critique assessment: 3
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15:52:58 4.2.62 (was 4.2.50). f(x)= 5x(x-1)^3. Give the zeros, the multiplicity of each, the behavior of the function near each zero and the large-|x| behavior of the function.
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RESPONSE --> The zeros are 0 with a multiplicity of 1, and 1 with a multiplicity of 3. The graph of the function resembles y = 5x^4 for large values of |x| I don't understand the question about the behavior of the function near each zero. confidence assessment: 1
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15:53:43 The zeros occur when x = 0 and when x - 1 = 0, so the zeros are at x = 0 (multiplicity 1) and x = 1 (multiplicity 3). Each zero is of odd degree so the graph crosses the x axis at each. If you multiply out all the terms you will be a polynomial with 5 x^4 as the highest-power term, i.e., the 'leading term'. For large | x | the polynomial resembles this 'leading term'--i.e., it resembles the power function 5 x^4.
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RESPONSE --> That's what it means... ok. So I take it ""behavior"" means oddness or evenness. self critique assessment: 3
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15:54:00 What is the maximum number of turning points on the graph of f?
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RESPONSE --> The maximum number of turning points are 3 confidence assessment: 2
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15:54:08 This is a polynomial of degree 4. A polynomial of degree n can have as many as n - 1 turning points. So this polynomial could possibly have as many as 4 - 1 = 3 turning points.
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RESPONSE --> ok self critique assessment: 3
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15:55:12 Give the intervals on which the graph of f is above and below the x-axis
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RESPONSE --> At the interval (-infinity, 0) the graph is above the x-axis, at the interval (0,1) the graph is below the x-axis, and at the interval (1, infinity) the graph is again above the x-axis. confidence assessment: 2
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15:55:36 this polynomial has zeros at x = 0 and x = 1. So on each of the intervals (-infinity, 0), (0, 1) and (1, infinity) the polynomial will lie either wholly above or wholly below the x axis. If x is a very large negative or positive number this fourth-degree polynomial will be positive, so on (-infinity, 0) and (1, infinity) the graph lies above the x axis. On (0, 1) we can test any point in this interval. Testing x = .5 we find that 5x ( x-1)^3 = 5 * .5 ( .5 - 1)^3 = -.00625, which is negative. So the graph lies below the x axis on the interval (0, 1).
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RESPONSE --> ok self critique assessment: 3
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