Answer :
Certainly! Here's a detailed, step-by-step solution for solving the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex]:
### Step 1: Factor out the common term
First, notice that [tex]\(x\)[/tex] is a common factor for all terms in the equation. We can factor out [tex]\(x\)[/tex] from the polynomial:
[tex]\[ x(x^3 + 9x^2 - 21x + 11) = 0 \][/tex]
### Step 2: Solve for the first factor
The first factor is [tex]\(x\)[/tex]. Solving [tex]\(x = 0\)[/tex] gives us one solution:
[tex]\[ x = 0 \][/tex]
### Step 3: Solve the remaining cubic equation
Now we need to solve the cubic polynomial:
[tex]\[ x^3 + 9x^2 - 21x + 11 = 0 \][/tex]
### Step 4: Use the Rational Root Theorem
The Rational Root Theorem suggests that possible rational roots of the polynomial are factors of the constant term (11) divided by factors of the leading coefficient (1). The factors of 11 are [tex]\( \pm1, \pm11 \)[/tex].
### Step 5: Test possible roots
Let's test these possible values to see if any of them are roots of the equation.
1. Test [tex]\( x = 1 \)[/tex]:
[tex]\[
(1)^3 + 9(1)^2 - 21(1) + 11 = 1 + 9 - 21 + 11 = 0
\][/tex]
Therefore, [tex]\( x = 1 \)[/tex] is a root.
2. Test [tex]\( x = -11 \)[/tex]:
[tex]\[
(-11)^3 + 9(-11)^2 - 21(-11) + 11 = -1331 + 1089 + 231 + 11 = 0
\][/tex]
Therefore, [tex]\( x = -11 \)[/tex] is a root.
### Step 6: Polynomial Division
Since [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex] are roots, to find any further roots, we can factor the polynomial using these values.
[tex]\[ x^3 + 9x^2 - 21x + 11 = (x - 1)(x + 11)(\text{another polynomial}) \][/tex]
Upon simplifying and verifying, we would see that we have factored the original polynomial into:
[tex]\[ (x)(x - 1)(x + 11) = 0 \][/tex]
### Step 7: Find all solutions
Therefore, the solutions to the original polynomial equation are:
[tex]\[ x = 0, x = 1, x = -11 \][/tex]
In conclusion, the solutions to the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex] are:
[tex]\[ x = 0, x = 1, x = -11 \][/tex]
### Step 1: Factor out the common term
First, notice that [tex]\(x\)[/tex] is a common factor for all terms in the equation. We can factor out [tex]\(x\)[/tex] from the polynomial:
[tex]\[ x(x^3 + 9x^2 - 21x + 11) = 0 \][/tex]
### Step 2: Solve for the first factor
The first factor is [tex]\(x\)[/tex]. Solving [tex]\(x = 0\)[/tex] gives us one solution:
[tex]\[ x = 0 \][/tex]
### Step 3: Solve the remaining cubic equation
Now we need to solve the cubic polynomial:
[tex]\[ x^3 + 9x^2 - 21x + 11 = 0 \][/tex]
### Step 4: Use the Rational Root Theorem
The Rational Root Theorem suggests that possible rational roots of the polynomial are factors of the constant term (11) divided by factors of the leading coefficient (1). The factors of 11 are [tex]\( \pm1, \pm11 \)[/tex].
### Step 5: Test possible roots
Let's test these possible values to see if any of them are roots of the equation.
1. Test [tex]\( x = 1 \)[/tex]:
[tex]\[
(1)^3 + 9(1)^2 - 21(1) + 11 = 1 + 9 - 21 + 11 = 0
\][/tex]
Therefore, [tex]\( x = 1 \)[/tex] is a root.
2. Test [tex]\( x = -11 \)[/tex]:
[tex]\[
(-11)^3 + 9(-11)^2 - 21(-11) + 11 = -1331 + 1089 + 231 + 11 = 0
\][/tex]
Therefore, [tex]\( x = -11 \)[/tex] is a root.
### Step 6: Polynomial Division
Since [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex] are roots, to find any further roots, we can factor the polynomial using these values.
[tex]\[ x^3 + 9x^2 - 21x + 11 = (x - 1)(x + 11)(\text{another polynomial}) \][/tex]
Upon simplifying and verifying, we would see that we have factored the original polynomial into:
[tex]\[ (x)(x - 1)(x + 11) = 0 \][/tex]
### Step 7: Find all solutions
Therefore, the solutions to the original polynomial equation are:
[tex]\[ x = 0, x = 1, x = -11 \][/tex]
In conclusion, the solutions to the equation [tex]\(x^4 + 9x^3 - 21x^2 + 11x = 0\)[/tex] are:
[tex]\[ x = 0, x = 1, x = -11 \][/tex]
