Question

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.

x^4-3x^3+19x^2+53x-174

Answer 1

To solve the given polynomial equation, we can use synthetic division to test for possible rational roots. By substituting potential rational roots into the equation and checking for zero results, we can identify the first root. In this case, the first root is x=2. To find the other roots, we divide the original equation by (x-2) using polynomial long division or synthetic division and solve the resulting equation of lower degree.

Step-by-step explanation:

To solve the polynomial equation, we can use synthetic division to test for possible rational roots. The equation is x^4-3x^3+19x^2+53x-174. We are looking for rational roots, so we can use the Rational Root Theorem to find possible values of x. The theorem states that any rational root must be a factor of the constant term (in this case, -174) divided by a factor of the leading coefficient (in this case, 1).

Using synthetic division, we can test each possible rational root by substituting it into the equation and checking if the result is equal to zero. The first root that satisfies this condition will be one of the solutions to the equation.

Let's start by listing the factors of the constant term (-174) and the factors of the leading coefficient (1):

• Factors of -174: -1, -2, -3, -6, -29, -58, 1, 2, 3, 6, 29, and 58
• Factors of 1: -1 and 1

By testing each possible rational root, we can determine that x=2 is one of the roots of the equation. To find the other roots, we can use polynomial long division or synthetic division to divide the original equation by (x-2). This will give us a new equation of degree 3, which we can then solve by factoring or using other methods.

Therefore, the first root of the equation x^4-3x^3+19x^2+53x-174 is x = 2.

Overall, the steps to solve the polynomial equation are:

1. List the factors of the constant term and the factors of the leading coefficient
2. Test each possible rational root using synthetic division
3. Find the first root that satisfies the condition (i.e., gives a result of zero)
4. Divide the original equation by the first root to obtain a new equation of lower degree
5. Solve the new equation to find additional roots