Polynomials Question (1 Viewer)

[Leffife]

Solve the equation x³ - 3x² - 4x + 12 = 0 given that the sum of two of its roots is zero.

 

[deswa1]

Let the roots be a,-a and b.
a+(-a)+b=3 ->b=3
a(-a)(3)=-12
a^2=4
a=2,-2

Therefore the solutions are x=2,-2,3

 

[Carrotsticks]

Solve the equation x³ - 3x² - 4x + 12 = 0 given that the sum of two of its roots is zero.

Let the roots be A, -A and B (since the sum of two of them is 0).

So sum of roots is B = 3

Product of roots is:



And so therefore we solve to find that A = plus/minus 2.

 

[Trebla]

Solve the equation x³ - 3x² - 4x + 12 = 0 given that the sum of two of its roots is zero.

Note this equation can be solved directly without the information about the sum of roots:
x³ - 3x² - 4x + 12 = 0
=> x²(x - 3) - 4(x - 3) = 0
=> (x² - 4)(x - 3) = 0
x = 3, 2, -2

 

[Leffife]

Hey thank you everyone!

Just one last question. I got stucked with.

Solve the equation 2x³ - 21x² + 42x - 16 = 0 , given that the roots form consecutive terms of a geometric sequence. Let the roots be alpha, alpha*beta , alpha*beta².

 

[Carrotsticks]

Hey thank you so much carrotstick.

Just one last question. I got stucked with.

Solve the equation 2x³ - 21x² + 42x - 16 = 0 , given that the roots form consecutive terms of a geometric sequence. Let the roots be alpha, alpha*beta , alpha*beta².

You probably should let the roots be:



Because if you use Beta, people may confuse it to be another root (since we often use alpha, beta, gamma etc to denote the roots).

Sum of roots (call this Equation 1):



Product of roots (call this Equation 2):



Making alpha the subject, we get:



Substitute into Equation 1 to acquire:



And so you can work out the roots by getting the value of alpha, and multiplying it by k once (for the second root) and then again for the third root.

 

[Carrotsticks]

Btw both yield the same solution.

For alpha = 8, k = 0.25, roots are:

8, 2, 0.5

For alpha = 0.5, k = 4, roots are:

0.5, 2, 8

ie: Exact same thing just different order.