For each of the sequences below, enter either diverges if the sequence diverges, or the limit of the sequence if the sequence co
1 answer:
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Answer:
- p(x) = x³ +0x² +0x +0
- p(x) = x³ +x² -x -1
- p(x) = x³ -x² -2x
Step-by-step explanation:
If the roots of p(x) are a, b, c, then it factors as ...
p(x) = (x -a)(x -b)(x -c)
and the expanded form is ...
p(x) = x^3 -(a+b+c)x^2 +(ac +b(a+c))x -abc
The requirement that the coefficients match the roots means we have three equations in three unknowns:
a = -(a+b+c)
b = ac +b(a+c)
c = -abc
The attachment shows the solutions to these equations. The solutions are found by making the substitutions ...
- b = x
- c = y
- a = -(x+y)/2
This leaves us with two equations in two unknowns that can be graphed on a Cartesian plane. The graph of the second equation is a circle:
x(1 -a -y) -ay = 0 ⇒ (x +1)^2 +y^2 = 1
The graph of the first equation is the union of the line y=0 and two hyperbolas. The points of intersection between this graph and the circle are ...
(x, y) = (-2, 0), (-1, -1), (0, 0) ⇒ (a, b, c) = (1, -2, 0), (1, -1, -1), (0, 0, 0)
The three polynomials that correspond to these values are shown above and in the attachment.
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<em>Additional comment</em>
There is an irrational fourth solution to the requirement that the roots match the coefficients.
Answer:
10 miles
Step-by-step explanation:
Given: Bike goes 8 miles south and then 6 miles east
To find: distance between the final point and the starting point
Solution:
<u>Pythagoras Theorem</u>
In a right angled triangle, square of hypotenuse is equal to the sum of squares of the two sides.
In ΔOAB,
Put
Take square root on both sides.
So, distance between the final point and the starting point is 10 miles