1) Find the roots of the polynomial equation. x^3-2x^2+10x+136=0
2) Use the rational root theorem to list all problem rational roots of the polynomial equation.
x^3+x^2-7x-4=0. Do not find the actual roots.
1) To find the roots of the polynomial equation x^3 - 2x^2 + 10x + 136 = 0, we can use the Rational Root Theorem and synthetic division.
Step 1: List all possible rational roots
The Rational Root Theorem states that the rational roots of a polynomial equation with integer coefficients are of the form p/q, where p is a factor of the constant term (136 in this case) and q is a factor of the leading coefficient (1 in this case).
The factors of 136 are ±1, ±2, ±4, ±8, ±17, ±34, ±68, ±136.
The factors of 1 are ±1.
So, the potential rational roots are ±1, ±2, ±4, ±8, ±17, ±34, ±68, ±136.
Step 2: Perform synthetic division
We can now perform synthetic division by trying these potential rational roots until we find one that gives a remainder of zero.
Let's start with x = 1:
1 | 1 -2 10 136
----------
-1, -3, 7, 143
The remainder is 143, so x = 1 is not a root.
Let's try x = -1:
-1 | 1 -2 10 136
----------
-1, 3, -13, 123
The remainder is 123, so x = -1 is not a root.
We need to continue this process with the remaining potential rational roots until we find one that gives a remainder of zero.
2) The Rational Root Theorem can be used to list potential rational roots of a polynomial equation without finding the actual roots.
To list all possible rational roots of the polynomial equation x^3 + x^2 - 7x - 4 = 0, we need to consider the factors of the constant term (-4) and the leading coefficient (1).
The factors of -4 are ±1, ±2, and ±4.
The factors of 1 are ±1.
Therefore, the potential rational roots are ±1, ±2, ±4.
Please note that listing the potential rational roots does not guarantee that they are the actual roots of the equation. To find the actual roots, you would need to use methods such as synthetic division or factoring.
a little synthetic division yields
(x+4)(x^2-6x+34) ...
well, what are the factors of 4?