Answer :
To solve the given equation [tex]\(x^6 - 4x^5 - 13x^4 + 28x^3 + 60x^2 = 0\)[/tex], given that [tex]\(x = -2\)[/tex] is a solution, we can use the following steps:
1. Recognize the Root: Since [tex]\(x = -2\)[/tex] is a solution to the equation, it means that the polynomial can be factored with [tex]\((x + 2)\)[/tex] as one of its factors.
2. Divide the Polynomial: To simplify the equation and find other solutions, we can perform polynomial division. We divide the entire polynomial [tex]\(x^6 - 4x^5 - 13x^4 + 28x^3 + 60x^2\)[/tex] by the factor [tex]\((x + 2)\)[/tex].
3. Determine the Result: After dividing the polynomial by [tex]\((x + 2)\)[/tex], we obtain a quotient polynomial, which in this case is:
[tex]\[
x^5 - 6x^4 - x^3 + 30x^2
\][/tex]
This means that the original polynomial can be expressed as:
[tex]\[
(x + 2)(x^5 - 6x^4 - x^3 + 30x^2) = 0
\][/tex]
4. Finding Other Roots: The remaining task is to find other solutions by setting the factored parts of the equation to zero:
[tex]\((x + 2) = 0\)[/tex] (which we know gives [tex]\(x = -2\)[/tex]) and
[tex]\(x^5 - 6x^4 - x^3 + 30x^2 = 0\)[/tex].
5. Solving the Factored Polynomial: The actual roots can be found by solving each part:
- We already have [tex]\(x = -2\)[/tex].
- To solve [tex]\(x^5 - 6x^4 - x^3 + 30x^2 = 0\)[/tex], you would typically use algebraic methods or numerical tools to find the remaining roots. This might involve factoring further or using synthetic division and the Rational Root Theorem to find integer solutions or employing numerical methods for non-integer roots.
By following these steps, you break down the problem into a simpler equation, initially reducing the degree from 6 to 5, and then further find the remaining roots step by step.
1. Recognize the Root: Since [tex]\(x = -2\)[/tex] is a solution to the equation, it means that the polynomial can be factored with [tex]\((x + 2)\)[/tex] as one of its factors.
2. Divide the Polynomial: To simplify the equation and find other solutions, we can perform polynomial division. We divide the entire polynomial [tex]\(x^6 - 4x^5 - 13x^4 + 28x^3 + 60x^2\)[/tex] by the factor [tex]\((x + 2)\)[/tex].
3. Determine the Result: After dividing the polynomial by [tex]\((x + 2)\)[/tex], we obtain a quotient polynomial, which in this case is:
[tex]\[
x^5 - 6x^4 - x^3 + 30x^2
\][/tex]
This means that the original polynomial can be expressed as:
[tex]\[
(x + 2)(x^5 - 6x^4 - x^3 + 30x^2) = 0
\][/tex]
4. Finding Other Roots: The remaining task is to find other solutions by setting the factored parts of the equation to zero:
[tex]\((x + 2) = 0\)[/tex] (which we know gives [tex]\(x = -2\)[/tex]) and
[tex]\(x^5 - 6x^4 - x^3 + 30x^2 = 0\)[/tex].
5. Solving the Factored Polynomial: The actual roots can be found by solving each part:
- We already have [tex]\(x = -2\)[/tex].
- To solve [tex]\(x^5 - 6x^4 - x^3 + 30x^2 = 0\)[/tex], you would typically use algebraic methods or numerical tools to find the remaining roots. This might involve factoring further or using synthetic division and the Rational Root Theorem to find integer solutions or employing numerical methods for non-integer roots.
By following these steps, you break down the problem into a simpler equation, initially reducing the degree from 6 to 5, and then further find the remaining roots step by step.
