School and Work

Use a graph and the Location Principle to find the real zeros of:

P(x) = 2x^3 - 4x + 1

1) (-1.3, 0.3, -1.5)
2) (1.3, -0.3, 1.5)
3) (-1.3, -0.3, 1.5)
4) (1.3, 0.3, 1.5)
5) (1.3, 0.3, -1.5)

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Factor-by-grouping yields:

-x³ - 4x + x^4 + 2x² - 8 = 0
-x(x² + 4) + (x² - 2)(x² + 4) = 0
(x² + 4)(x² - 2 - x) = 0
(x² + 4)(x² - x - 2) = 0
(x² + 4)(x - 2)(x + 1) = 0

Finally, by zero-product property:

x² + 4 = 0 and x - 2 = 0 and x + 1 = 0
x² = -4 and x = {2,-1}
x = ±v-4 and x = {-1,2}
x = {-1, 2, ±2i}

I hope this helps!

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a. use newtons method to find all the values x such that x^2 - 2v2x+1 = 0 (the 2x + 1 is under the radical).

b. Use Newton's Method to approximate the positive solution to the equation ln(x + 4) = x to three
decimal places.

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