The smallest prime number of p for which p^3 + 4p^2 + 4p has exactly 30 positive divisors is 43.
<h3>What is the smallest prime number of p for which p must have exactly 30 positive divisors?</h3>
The smallest number of p in the polynomial equation p^3 + 4p^2 + 4p for which p must have exactly 30 divisors can be determined by factoring the polynomial expression, then equating it to the value of 30.
i.e.
By factorization, we have:
Now, to get exactly 30 divisor.
- (p+2)² requires to give us 15 factors.
Therefore, we can have an equation p + 2 = p₁ × p₂²
where:
- p₁ and p₂ relate to different values of odd prime numbers.
So, for the least values of p + 2, Let us assume that:
p + 2 = 5 × 3²
p + 2 = 5 × 9
p + 2 = 45
p = 45 - 2
p = 43
Therefore, we can conclude that the smallest prime number p such that
p^3 + 4p^2 + 4p has exactly 30 positive divisors is 43.
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Represent these consecutive numbers (assuming that they are all integers):
x
x+1
x+2
x+3
x+4
x+5
and so on
x+8
x+9 is the tenth number. x+9 = 10, so x = 9.
Think of it this way: there are 10 consecutive numbers, and the last one is 10.
Working backwards, we get the sequence 10, 9, ... 3, 2, 1.
The sum of such an arith sequence is equal to the count of the numbers times the average of the first and last terms:
sum here = 10(1+10)/2 = 5(11) = 55 (answer)
Good question I’m not sure what it is but I’m sure you can look on google
Answer:
Equation of line in slope-intercept form is: y = -8x-39
Step-by-step explanation:
Given points are:
(-4, -7) and (-6, 9)
The slope-intercept form is given by the equation
Here m is the slope of the line and b is the y-intercept.
m is found using the formula
Here
(x1,y1) = (-4-7)
(x2,y2) = (-6,9)
Putting the values in the formula
Putting slope in general equation
Putting (-6,9) in the equation
Putting the value of b we get
Hence,
Equation of line in slope-intercept form is: y = -8x-39
Answer:
Step-by-step explanation:
The standard form of an equation of a circle:
(h, k) - center
r - radius
We have the endpoints of the diameter of a circle (-8, -6) and (-4, -14).
The midpoint of a diameter is a center of a circle.
The formula of a midpoint:
Substitute:
We have h = -6 and k = -10.
The radius is the distance between a center and the point on a circumference of a circle.
The formula of a distance between two points:
Substitute (-6, -10) and (-8, -6):
Finally we have
