Precal Help?? Rational and Irrational Roots????

Question

The equation has exactly one positive root. Locate the root between successive hundredths. Determine the successive integer bounds by computing f(0), f(1), f(2), and so on, until you find a sign change.

x³ − 8x² + 21x − 37 = 0

lower bound
upper bound

 

Answer 1

The function  is f(x) = x³ - 8x² + 21x - 37 = 0.

Use the method of successive approximations to locate this root between successive hundredths.

Table

x       -3        -2       -1           0  .............       4          5         6     7

f(x)   -199   -119     -67        -37 ............. -17           -7         59      61

                                                                         (Sign change)

The sign change is a signal that the function has real root between 5 and 6.

Graph

From the figure the function real root between 5 and 6.

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  = 5.37129

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

Since f (5.37129) = 0, x = 5.37129 is a zero.

The depressed polynomial is

Since the depressed polynomial of this zero, , is quadratic,

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

x³ - 8x² + 21x - 37 = 0 have two imaginary roots.

Therefore, positive real root x = 5.37129.