Answer :
To solve the polynomial equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that satisfy this equation. Here's a detailed step-by-step guide.
1. Factor Out Common Terms:
Start by factoring out the common term, [tex]\(x\)[/tex], from the entire equation. This simplifies the equation significantly:
[tex]\[
x(x^3 + 9x^2 - 21x + 11) = 0
\][/tex]
This implies one solution is [tex]\(x = 0\)[/tex].
2. Solve the Cubic Equation:
We are left with the cubic equation [tex]\(x^3 + 9x^2 - 21x + 11 = 0\)[/tex]. To find its roots, we can either use methods like synthetic division or check for possible rational roots using the Rational Root Theorem. The Rational Root Theorem suggests that the possible rational roots could be factors of the constant term, 11, divided by the factors of the leading coefficient, 1. Thus, we consider [tex]\( \pm 1, \pm 11 \)[/tex].
3. Evaluate Possible Roots:
- Checking [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.
4. Use Synthetic Division:
With [tex]\(x = 1\)[/tex] as a confirmed root, we can divide the cubic polynomial by [tex]\(x - 1\)[/tex] using synthetic division to find the remaining factors.
- Divide [tex]\(x^3 + 9x^2 - 21x + 11\)[/tex] by [tex]\(x-1\)[/tex].
- This yields [tex]\(x^2 + 10x - 11\)[/tex].
5. Factor the Quadratic:
Next, solve [tex]\(x^2 + 10x - 11 = 0\)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = -11\)[/tex].
[tex]\[
x = \frac{-10 \pm \sqrt{10^2 - 4 \times 1 \times (-11)}}{2 \times 1}
\][/tex]
[tex]\[
x = \frac{-10 \pm \sqrt{100 + 44}}{2}
\][/tex]
[tex]\[
x = \frac{-10 \pm \sqrt{144}}{2}
\][/tex]
[tex]\[
x = \frac{-10 \pm 12}{2}
\][/tex]
This gives the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex].
6. Compile the Solutions:
The solutions to the original polynomial equation are:
[tex]\[
x = 0, x = 1, x = -11
\][/tex]
Thus, the solutions for 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].
1. Factor Out Common Terms:
Start by factoring out the common term, [tex]\(x\)[/tex], from the entire equation. This simplifies the equation significantly:
[tex]\[
x(x^3 + 9x^2 - 21x + 11) = 0
\][/tex]
This implies one solution is [tex]\(x = 0\)[/tex].
2. Solve the Cubic Equation:
We are left with the cubic equation [tex]\(x^3 + 9x^2 - 21x + 11 = 0\)[/tex]. To find its roots, we can either use methods like synthetic division or check for possible rational roots using the Rational Root Theorem. The Rational Root Theorem suggests that the possible rational roots could be factors of the constant term, 11, divided by the factors of the leading coefficient, 1. Thus, we consider [tex]\( \pm 1, \pm 11 \)[/tex].
3. Evaluate Possible Roots:
- Checking [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.
4. Use Synthetic Division:
With [tex]\(x = 1\)[/tex] as a confirmed root, we can divide the cubic polynomial by [tex]\(x - 1\)[/tex] using synthetic division to find the remaining factors.
- Divide [tex]\(x^3 + 9x^2 - 21x + 11\)[/tex] by [tex]\(x-1\)[/tex].
- This yields [tex]\(x^2 + 10x - 11\)[/tex].
5. Factor the Quadratic:
Next, solve [tex]\(x^2 + 10x - 11 = 0\)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = -11\)[/tex].
[tex]\[
x = \frac{-10 \pm \sqrt{10^2 - 4 \times 1 \times (-11)}}{2 \times 1}
\][/tex]
[tex]\[
x = \frac{-10 \pm \sqrt{100 + 44}}{2}
\][/tex]
[tex]\[
x = \frac{-10 \pm \sqrt{144}}{2}
\][/tex]
[tex]\[
x = \frac{-10 \pm 12}{2}
\][/tex]
This gives the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex].
6. Compile the Solutions:
The solutions to the original polynomial equation are:
[tex]\[
x = 0, x = 1, x = -11
\][/tex]
Thus, the solutions for 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].
