1) Find the roots of the polynomial equation. x^3-2x^2+10x+136=0
To find the roots of the polynomial equation x^3 - 2x^2 + 10x + 136 = 0, we can use various methods such as factoring, synthetic division, or graphing. One way to approach this is by using factoring through Rational Root Theorem.
Step 1: Rational Root Theorem
The Rational Root Theorem states that if a polynomial equation has a rational root (in the form p/q), then p must be a factor of the constant term (136) and q must be a factor of the leading coefficient (1).
The factors of 136 are ±1, ±2, ±4, ±8, ±17, ±34, ±68, and ±136.
The factors of 1 (leading coefficient) are ±1.
Possible rational roots: ±1, ±2, ±4, ±8, ±17, ±34, ±68, and ±136.
Step 2: Synthetic Division
Now, we can perform synthetic division using the possible rational roots to determine if any of them are indeed roots of the equation.
Let's try x = 1:
1 | 1 -2 10 136
| 1 -1 9
________________
1 -1 9 145
Since the remainder is not 0, x = 1 is not a root.
Let's try x = -1:
-1 | 1 -2 10 136
| -1 3 -13
________________
1 -3 13 123
Again, the remainder is not 0, so x = -1 is not a root.
Continuing the process for the rest of the possible rational roots, we find that none of them are roots of the equation.
Thus, the polynomial equation x^3 - 2x^2 + 10x + 136 = 0 does not have any rational roots. The roots may be irrational or complex numbers.