1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
andrezito [222]
3 years ago
9

Suppose the roots of the polynomial $x^2 - mx + n$ are positive prime integers (not necessarily distinct). Given that $m < 20

,$ how many possible values of $n$ are there?
Mathematics
1 answer:
Vsevolod [243]3 years ago
3 0

Answer:

<em>18</em> values for n are possible.

Step-by-step explanation:

Given the quadratic polynomial:

$x^2 - mx + n$

such that:

Roots are positive prime integers and

$m < 20$

To find:

How many possible values of n are there ?

Solution:

First of all, let us have a look at the sum and product of a quadratic equation.

If the quadratic equation is:

Ax^{2} +Bx+C

and the roots are: \alpha and \beta

Then sum of roots, \alpha+\beta = -\frac{B}{A}

Product of roots, \alpha \beta = \frac{C}{A}

Comparing the given equation with standard equation, we get:

A = 1, B = -m and C = n

Sum of roots,  \alpha+\beta = -\frac{-m}{1} = m

Product of roots, \alpha \beta = \frac{n}{1} = n

We are given that m  

\alpha and \beta are positive prime integers such that their sum is less than 20.

Let us have a look at some of the positive prime integers:

2, 3, 5, 7, 11, 13, 17, 23, 29, .....

Now, we have to choose two such prime integers from above list such that their sum is less than 20 and the roots can be repetitive as well.

So, possible combinations and possible value of n (= \alpha \times \beta) are:

1.\ 2,  2\Rightarrow  n = 2\times 2 = 4\\2.\ 2, 3 \Rightarrow  n = 6\\3.\ 2, 5 \Rightarrow  n = 10\\4.\ 2,  7\Rightarrow  n = 14\\5.\ 2, 11 \Rightarrow  n = 22\\6.\ 2, 13 \Rightarrow  n = 26\\7.\ 2, 17 \Rightarrow  n = 34\\8.\ 3,  3\Rightarrow  n = 3\times 3 = 9\\9.\ 3, 5 \Rightarrow  n = 15\\10.\ 3, 7 \Rightarrow  n = 21\\

11.\ 3,  11\Rightarrow  n = 33\\12.\ 3, 13 \Rightarrow  n = 39\\13.\ 5, 5 \Rightarrow  n = 25\\14.\ 5, 7 \Rightarrow  n = 35\\15.\ 5, 11 \Rightarrow  n = 55\\16.\ 5, 13 \Rightarrow  n = 65\\17.\ 7, 7 \Rightarrow  n = 49\\18.\ 7, 11 \Rightarrow  n = 77

So,as shown above <em>18 values for n are possible.</em>

You might be interested in
What is the period of the function f(x)=sin(x/2)
Liula [17]
The period of the function is 2pi
3 0
3 years ago
What is the tan invers of 3i/-1-i​
postnew [5]

<em>z</em> = 3<em>i</em> / (-1 - <em>i</em> )

<em>z</em> = 3<em>i</em> / (-1 - <em>i</em> ) × (-1 + <em>i</em> ) / (-1 + <em>i</em> )

<em>z</em> = (3<em>i</em> × (-1 + <em>i</em> )) / ((-1)² - <em>i</em> ²)

<em>z</em> = (-3<em>i</em> + 3<em>i</em> ²) / ((-1)² - <em>i</em> ²)

<em>z</em> = (-3 - 3<em>i </em>) / (1 - (-1))

<em>z</em> = (-3 - 3<em>i </em>) / 2

Note that this number lies in the third quadrant of the complex plane, where both Re(<em>z</em>) and Im(<em>z</em>) are negative. But arctan only returns angles between -<em>π</em>/2 and <em>π</em>/2. So we have

arg(<em>z</em>) = arctan((-3/2)/(-3/2)) - <em>π</em>

arg(<em>z</em>) = arctan(1) - <em>π</em>

arg(<em>z</em>) = <em>π</em>/4 - <em>π</em>

arg(<em>z</em>) = -3<em>π</em>/4

where I'm taking arg(<em>z</em>) to have a range of -<em>π</em> < arg(<em>z</em>) ≤ <em>π</em>.

6 0
2 years ago
Write expression without using exponents.<br><br> X^2
wariber [46]

Answer:

x(x) or x*x

Step-by-step explanation:

To write this in the expanded form you just multiply the base(x) by itself

6 0
3 years ago
I REALLY NEED HELP, ILL MEDAL!!!!
lara31 [8.8K]
Well, first of all, the first statement  (ABC = ADC) looks like it just says
that the two halves of the little square ... each side of the diagonal ...
are congruent.  That's no big deal, and it's no help in answering the
question.

The effect of the dilation is that all the DIMENSIONS of the square
are doubled ... each side of the square becomes twice as long.

Then, when you multiply (length x width) to get the area, you'd have

           Area = (2 x original length) x (2 x original width)

and that's
the same as      (2 x 2) x (original length x original width)

                        =     (4)  x  (original area) .


Here's an easy, useful factoid to memorize:

-- Dilate a line (1 dimension) by 'x' times . . . multiply the length by  x¹

-- Dilate a shape (2 dimensions) by 'x' . . . multiply area by  x²

-- Dilate a solid (3 dimensions) by 'x' . . . multiply volume by  x³

And that's all the dimensions we have in our world.
_______________________________

Oh, BTW . . .

-- Dilate a point (0 dimensions) by 'x' . . . multiply it by  x⁰  (1)

5 0
2 years ago
Get moneyy. Answer this.
Alexxandr [17]
It should be a 80 degrees.
5 0
2 years ago
Read 2 more answers
Other questions:
  • Evan has a summer job to pick berries on a farm.He earns $2.00 every 15 minutes that he picks strawberries.He earns $2.40 for ev
    9·1 answer
  • 6) The sum of three consecutive numbers is seventy-eight.
    11·2 answers
  • Factor completely 9xy + 36x − 10y − 40.
    11·2 answers
  • Help me I need help!
    9·1 answer
  • Point-Slope Form of a Line
    10·1 answer
  • A certain group of test subjects had pulse rates with a mean of 75.2 beats per minute and a standard deviation of 11.2 beats per
    12·1 answer
  • Please help!!! I’ll mark as brainliest :)
    5·1 answer
  • At a sale, dresses were sold for 36% of their original price. If the dresses originally cost $200 each, how much did a dress cos
    5·1 answer
  • Multiply and simplify.
    15·1 answer
  • Make a list of all the different possible outcomes when two number cubes are rolled. What fraction of the possible outcomes is f
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!