Answer :
To solve the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that make this equation true. Let's solve it step by step.
1. Factor Out the Common Term:
Notice that each term in the equation has [tex]\(x\)[/tex] as a factor. So, we can factor out [tex]\(x\)[/tex] from the entire equation:
[tex]\[
x(x^3 + 9x^2 - 21x + 11) = 0
\][/tex]
This gives us one solution directly from the factor [tex]\(x = 0\)[/tex].
2. Solve the Cubic Equation:
Now, we need to solve the cubic polynomial [tex]\(x^3 + 9x^2 - 21x + 11 = 0\)[/tex].
3. Possible Rational Roots:
For the cubic equation, we check for possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (11) divided by the factors of the leading coefficient (1). Hence, the possible rational roots are ±1, ±11.
4. Test the Possible Roots:
- Evaluate the polynomial at [tex]\(x = 1\)[/tex]:
[tex]\[
1^3 + 9 \times 1^2 - 21 \times 1 + 11 = 1 + 9 - 21 + 11 = 0
\][/tex]
So, [tex]\(x = 1\)[/tex] is a root.
5. Factor the Cubic Polynomial:
Since [tex]\(x = 1\)[/tex] is a root, we can factor the cubic polynomial as [tex]\((x - 1)\)[/tex] times another polynomial:
[tex]\[
x^3 + 9x^2 - 21x + 11 = (x - 1)(x^2 + bx + c)
\][/tex]
Use synthetic division or polynomial long division to divide the cubic polynomial by [tex]\((x - 1)\)[/tex] to find the quadratic polynomial [tex]\((x^2 + bx + c)\)[/tex].
6. Continue Factoring or Solving:
The quadratic can be tested for further factorization. If it's factorizable over the rationals, factor it. Otherwise, use the quadratic formula to find the remaining roots.
Solving or factoring further, we find:
[tex]\[
(x - 1)(x + 11)
\][/tex]
7. Final Solutions:
Therefore, the roots of the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex] are [tex]\(x = 0\)[/tex], [tex]\(x = 1\)[/tex], and [tex]\(x = -11\)[/tex].
These steps outline a logical process to find the roots of the given polynomial equation.
1. Factor Out the Common Term:
Notice that each term in the equation has [tex]\(x\)[/tex] as a factor. So, we can factor out [tex]\(x\)[/tex] from the entire equation:
[tex]\[
x(x^3 + 9x^2 - 21x + 11) = 0
\][/tex]
This gives us one solution directly from the factor [tex]\(x = 0\)[/tex].
2. Solve the Cubic Equation:
Now, we need to solve the cubic polynomial [tex]\(x^3 + 9x^2 - 21x + 11 = 0\)[/tex].
3. Possible Rational Roots:
For the cubic equation, we check for possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (11) divided by the factors of the leading coefficient (1). Hence, the possible rational roots are ±1, ±11.
4. Test the Possible Roots:
- Evaluate the polynomial at [tex]\(x = 1\)[/tex]:
[tex]\[
1^3 + 9 \times 1^2 - 21 \times 1 + 11 = 1 + 9 - 21 + 11 = 0
\][/tex]
So, [tex]\(x = 1\)[/tex] is a root.
5. Factor the Cubic Polynomial:
Since [tex]\(x = 1\)[/tex] is a root, we can factor the cubic polynomial as [tex]\((x - 1)\)[/tex] times another polynomial:
[tex]\[
x^3 + 9x^2 - 21x + 11 = (x - 1)(x^2 + bx + c)
\][/tex]
Use synthetic division or polynomial long division to divide the cubic polynomial by [tex]\((x - 1)\)[/tex] to find the quadratic polynomial [tex]\((x^2 + bx + c)\)[/tex].
6. Continue Factoring or Solving:
The quadratic can be tested for further factorization. If it's factorizable over the rationals, factor it. Otherwise, use the quadratic formula to find the remaining roots.
Solving or factoring further, we find:
[tex]\[
(x - 1)(x + 11)
\][/tex]
7. Final Solutions:
Therefore, the roots of the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex] are [tex]\(x = 0\)[/tex], [tex]\(x = 1\)[/tex], and [tex]\(x = -11\)[/tex].
These steps outline a logical process to find the roots of the given polynomial equation.
