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cestrela7 [59]
3 years ago
12

Prove that x-1 is a factor of x^n-1 for any positive integer n.

Mathematics
1 answer:
PolarNik [594]3 years ago
3 0

Answer:    

x-1 is a factor of x^n - 1

Step-by-step explanation:

x-1 is a factor of x^n - 1

We will prove this with the help of principal of mathematical induction.

For n = 1, x-1 is a factor x-1, which is true.

Let the given statement be true for n = k that is x-1 is a factor of x^k - 1.

Thus, x^k - 1 can be written equal to  y(x-1), where y is an integer.

Now, we will prove that the given statement is true for n = k+1

x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)

Thus, x^k - 1 is divisible by x-1.

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.

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The triangular pyramid shown is made up of four congruent triangles. The length of the base is 8 inches, and the height of the b
Aneli [31]

Answer:

(B) Talia is correct. The lateral area can be found by approximating one large triangle, which can be found using the expression 4 (one-half (8) (6.9))

Step-by-step explanation:

Base of the Pyramid = 8 Inches

Height of the Triangular Face = 6.9 Inches

In any solid shape, the Lateral surface area is the sum of all sides except its top and bottom bases.

Since the four triangles are congruent:

Lateral Surface Area = 4 X Area of One Triangle

Area of a Triangle = \frac{1}{2}$ X Base X Height

Area of one Triangular Face = \dfrac{1}{2}(8)(6.9)

Therefore:

Lateral Surface Area = 4\left(\dfrac{1}{2}(8)(6.9)\right)

Therefore, Talia is correct.

8 0
3 years ago
Seattle, Washington is known for being very rainy. One day last month, 8 inches of rain fell in 1 1/2 hours. What is the rate of
Leno4ka [110]

Answer:

0.44ft/hr

Step-by-step explanation:

Given parameters:

Quantity of rainfall  = 8inches

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   so;

            \frac{8}{12}   = 0.67ft

Time  = \frac{3}{2}hr

Unknown:

Rate of rainfall expressed in ft/hr  = ?

Solution:

The rate of the rainfall is given as;

    Rate  = \frac{Quantity}{time}  

  Rate  = \frac{0.67ft}{\frac{3}{2} }    = 0.44ft/hr

8 0
3 years ago
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Calvin is painting an area that is 115 square feet. He is able to paint 2.5 square feet every 2 minutes. At this rate , how long
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Answer:

Calvin will paint the entire area in 92 minutes or 1 hour 32 minutes.

Step-by-step explanation:

First of all we know that the total surface are that Calvin is painting is 115 square feet.

The rate at which Calvin is painting refers to the surface area he covers every 2 minutes. This is 2.5 square feet every 2 minutes.

we can set up a relation to solve this problem

If Calvin paints 2.5 ft^{2} in 2 minutes

He will paint 115 ft^{2} in x minutes

By cross multiplying, we have x= \frac{2 \times 115}{2.5}=92 minutes

Hence Calvin will paint the entire area in 92 minutes or 1 hour 32 minutes.

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