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3 years ago

X^3 + 5x^2 - 8x^2 - 40x + 7x + 35

Mathematics

1 answer:

Temka [501]3 years ago

3 0

Remark

x - 1 is a root.

Proof

x - 1 = 0

x = 1

(1)^3 + 5(1)^2 - 8(1)^2 - 40(1) + 7(1) + 35

1 + 5 - 8 - 40 + 7 + 35

6 + 7 + 35 - 48 = 0

Solution

Divide x - 1 into the original equation.

x^3 + 5x^2 - 8x^2 - 33x + 35

x - 1 || x^3 - 3x^2 - 33x + 35 || x^2 - 2x - 35

x^3 - x^2

-2x^2 - 33x

-2x^2 + 2x

- 35x + 35

-35x + 35

x^2 - 2x - 35 factors into (x - 7)(x + 5)

Complete factorization

(x - 1)(x - 7)(x + 5)

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For this case we have the following function given:

$f(t) = s = t^3 -9t^2 +15 t$

Part a: Find the velocity at time t.

For this case we just need to take the derivate of the position function respect to t like this:

$\frac{ds}{dt}= v(t) = 3t^2 -18t +15$

Part b: What is the velocity after 3 s?

For this case we just need to replace t=3 s into the velocity equation and we got:

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Part c: When is the particle at rest?

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$3t^2 -18 t +15 =0$

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Part d: When is the particle moving in the positive direction? (Enter your answer in interval notation.)

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So then the intervals positive are $[0,1) \cup (5,\infty)$

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