Roots of

Question

F(x)=(4x^3)-(6x^2)+9x+10?

Answer 1

The cubic polynomial is 4x ³ - 6x ² + 9x + 10.

Graph :

The points where it crosses the x  axis  will give solutions to the polynimial function .

The graph crosses the x  - axis at a point that would suggest a factor.

It crosses the x  - axis at one point hence there are one real root.

x  = - 0.67

Use synthatic division to detrmine if the given value of x  is a root of the polynomial.

- 0.67  | 4   -6       9    10

          | 0   -2.7    6    -10       

          _________________

          4   -8.7     15    |0

Since f (- 0.67) = 0, x = - 0.67 is a root to the given polynomial..

The depressed polynomial is  4x ²- 8.7x + 15 = 0.

Since the depressed polynomial of this zero, 4x ²- 8.7x + 15 = 0, is a quadratic,

Use the Quadratic Formula to find the roots of the related quadratic equation

 

a = 4, b = - 8.7, and c = 15.

x = [ 8.7 ± √(- 8.7)² - 4 * 4 * 15) ] / 2 * 4

x = [ 8.7 ± √75.69 - 240) ] / 8

x = [ 8.7 ± √(- 164.31)) ] / 8

Substitute i ²  for - 1.

x = [ 8.7 ± √(i ²164.31)) ] / 8

x = [ 8.7 ± 12.82i ] / 8

Real root is x = - 0.67.

And complex roots are x = [ 8.7 - 12.82i ] / 8 and x = [ 8.7 + 12.82i ] / 8.