# Finding $|k|$ of the Polynomial $x^3-kx+25$

• MHB
• anemone
In summary, the polynomial $x^3-kx+25$ has three real roots, and two of these roots sum to 5. Using Vieta's formula, we can find that the remaining root is -5 and that k = 20. Although the assumption that the other two roots are complex leads to a contradiction, the only possible value for k is still 20. Therefore, the solution for k is 20.
anemone
Gold Member
MHB
POTW Director
The polynomial $x^3-kx+25$ has three real roots. Two of these root sum to 5. What is $|k|$?

[sp]
Let the roots be a, b, and c.

Then by Vieta's formula
a + b + c = 0

Take b + c = 5. Then a = -5,

So we know that
$$\displaystyle a^3 - k a + 25 = 0$$

$$\displaystyle (-5)^3 - k ( -5 ) + 25 = 0$$

k = 20

Note that we haven't used the condition that all roots are real, so let's assume that b and c are complex and that this gives a contradiction.

We know that a = -5 and that b and c are, by assumption, complex. So let b = m + in and c = 5 - (m + in).

The other Vieta formula says that
abc = 25

$$\displaystyle (-5) ( m + in) ( 5 - (m + in) ) = 25$$

$$\displaystyle (-m^2 + 5m + n^2) + (5n - 2mn)i = -5$$

Using the second term
$$\displaystyle 5n - 2mn = 0$$

So n = 0 or m = 5/2.

Let m = 5/2 and put it into the first term:
$$\displaystyle -m^2 + 5m + n^2 = -5$$

$$\displaystyle - \left ( \dfrac{5}{2} \right ) ^2 + 5 \left ( \dfrac{5}{2} \right ) + n^2 = -5$$

$$\displaystyle n^2 = -\dfrac{45}{4}$$
which says that $$\displaystyle n^2 < 0$$, which is impossible.

Thus n = 0 and the solution for k = 20 stands as the only possible k.
[/sp]
-Dan

Above is good method and here is my
as $x^2$ term is zero and sum of 2 roots is 5 so $3^{rd}$ root is -5 so we have the product of 2 roots = 5 (as product of all roots -25)

$x^3-xk+25 = (x+5)(x^2- 5x + 5) = x^3 -5x^2+5x + 5x^2 - 25 x + 25 = x^3 -20x + 25$

comparing with given equation we have k = 20 and hence $| k | = 20$

we have to check that $x^2-5x+5=0$ is having real roots or not. We have discriminant = $5^2-20= 5 >0$ so real roots

so ans 20

Last edited:
[sp]
There's a typo.

$$\displaystyle x^3 - k x + 25 = x^3 - 20 x + 25 \implies k = 20$$, not -20.
[/sp]
-Dan

topsquark said:
[sp]
There's a typo.

$$\displaystyle x^3 - k x + 25 = x^3 - 20 x + 25 \implies k = 20$$, not -20.
[/sp]
-Dan
Thanks I have done the needful in line for the flow

## 1. What is the degree of the polynomial?

The degree of a polynomial is the highest exponent of the variable. In this case, the degree is 3 because the highest exponent of x is 3.

## 2. How do you find the value of k?

To find the value of k, we can use the rational root theorem or synthetic division to test different values of k until we find one that satisfies the equation. We can also use a graphing calculator to find the x-intercepts of the polynomial, which will give us the value of k.

## 3. Can the polynomial have more than one value of k?

Yes, the polynomial can have more than one value of k that satisfies the equation. This is because a polynomial can have multiple roots or solutions.

## 4. How does the value of k affect the graph of the polynomial?

The value of k affects the graph of the polynomial by shifting it horizontally. If k is positive, the graph will shift to the right, and if k is negative, the graph will shift to the left. The value of k also affects the number of roots or solutions of the polynomial.

## 5. What is the significance of finding the value of k?

Finding the value of k allows us to fully understand the behavior and characteristics of the polynomial. It helps us determine the number of roots, the direction of the graph, and the behavior of the polynomial at certain points. This information is useful in solving real-world problems and making predictions based on the polynomial's behavior.

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