Question 5 Multiple Choice Worth 1 points)

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

Vsevolod [243]

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>

3 0

3 years ago

(2x + 5x - 3) + (x + 3)

IrinaK [193]

Answer:

8x

Step-by-step explanation:

Because you are adding the (x+3) there you can immediately you the association property to simplify

(2x + 5x + x) + (3 - 3)

8x + 0 =

8x

5 0

2 years ago

What is the total amount of fees if there were 6 withdrawals for cash in a month, 10 uses of an ATM,

almond37 [142]

Answer:

34

Step-by-step explanation:

16+10+6=34

3 0

2 years ago

An expression is shown below: f(x) = 2x2 − 3x − 5 Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 poi

skelet666 [1.2K]

Answer:

Step-by-step explanation:

I'm going to assume you meant 2x²-3x-5

8 0

3 years ago

Riley collected rain for several weeks in a rain gauge he collected 1.48 in the first week half that amount the 2nd weekhow much

Lostsunrise [7]

2.79 in. Half of 1.48 is .64 and if you add them you come to 2.21 then subtract 2.21 from 5 and you get 2.79.

3 0

3 years ago