Question

Let r and c be real numbers. If r and −r are roots of 5x3+cx2−10x+9=0,then c equals

• A. \(\frac{-9}{2}\)
• B. \(\frac{9}{2}\)
• C. −4
• D. 4

Answer 1

The Correct Option is A

The given polynomial equation is 5x³ + cx² - 10x + 9 = 0.
We are told that r and -r are roots of this equation, which means that if we substitute these values into the equation, it should become zero:

For r:
5r³ + cr² - 10r + 9 = 0
For -r:
5(-r)³ + c(-r)² - 10(-r) + 9 = 0
-5r³ + cr² + 10r + 9 = 0
Since r and -r are roots, both of these equations must be true. Now, let's add the two equations:

(5r³ + cr² - 10r + 9) + (-5r³ + cr² + 10r + 9) = 0
cr² + cr² + 18 = 0
2cr² + 18 = 0

Now, divide both sides of the equation by 2r² (as long as r ≠ 0, since division by zero is undefined):
\(c + \frac{9}{r}^2 = 0\)

Since we are given that r ≠ 0, we can multiply both sides of the equation by r^2 to isolate c:

\(c = -\frac{9}{r}^2\)

Now, let's consider the options:

a) -(9/2)
b) 4
c) -4
d) 9/2

Comparing these options with the derived equation \(c = -\frac{9}{r}^2\), we can see that the correct answer is indeed option 'a' -(9/2), as it matches the value of c that we obtained.