Question

Find the complex roots and possible number of real and imaginary roots for the equations. Then find all roots . Show Steps to Solve.

F(x)=x³+2x²-8x
And
F(x)=x³+5x²+6x

Answer 1

The first equation has two real roots and one complex root, while the second equation has three real roots and no complex roots.

Step-by-step explanation:

The given equations are:

F(x) = x³ + 2x² - 8x

F(x) = x³ + 5x² + 6x

Equation F(x) = x³ + 2x² - 8x:

To find the complex roots, we can use the fact that if a polynomial equation has complex roots, they will always occur in conjugate pairs. Let's factor the equation:

F(x) = x(x² + 2x - 8)

F(x) = x(x + 4)(x - 2)

From this factorization, we can see that the equation has two real roots, x = -4 and x = 2, and one complex root, x = 0. Therefore, there are two real roots and one complex root.

Equation F(x) = x³ + 5x² + 6x:

Let's factor the equation:

F(x) = x(x² + 5x + 6)

F(x) = x(x + 2)(x + 3)

The factorization shows that the equation has three real roots, x = 0, x = -2, and x = -3. Therefore, there are three real roots and no complex roots.