Question
Find all zeros of f(x)=x^4-x^3-20x^2+46x-20. Enter the zeros separated by commas. Enter exact values, not decimal approximations.
Answers
GPT-4o mini
To find the zeros of the polynomial \( f(x) = x^4 - x^3 - 20x^2 + 46x - 20 \), we can start by attempting to factor the polynomial or use the Rational Root Theorem to find potential rational roots.
### Step 1: Apply the Rational Root Theorem
The possible rational roots are the factors of the constant term (-20) divided by the factors of the leading coefficient (1). This gives us the possible rational roots:
\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20
\]
### Step 2: Test Possible Rational Roots
We'll test these candidates in the polynomial \( f(x) \).
- Testing \( x = 1 \):
\[
f(1) = 1^4 - 1^3 - 20(1^2) + 46(1) - 20 = 1 - 1 - 20 + 46 - 20 = 6 \quad (\text{not a root})
\]
- Testing \( x = 2 \):
\[
f(2) = 2^4 - 2^3 - 20(2^2) + 46(2) - 20 = 16 - 8 - 80 + 92 - 20 = 0 \quad (\text{is a root})
\]
Since \( x = 2 \) is a root, we can divide \( f(x) \) by \( x - 2 \).
### Step 3: Synthetic Division by \( x - 2 \)
\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & -20 & 46 & -20 \\
& & 2 & 2 & -36 & 20 \\
\hline
& 1 & 1 & -18 & 10 & 0 \\
\end{array}
\]
This gives us \( f(x) = (x - 2)(x^3 + x^2 - 18x + 10) \).
### Step 4: Find the Roots of \( x^3 + x^2 - 18x + 10 \)
We will apply the Rational Root Theorem again for \( g(x) = x^3 + x^2 - 18x + 10 \) with possible roots \( \pm 1, \pm 2, \pm 5, \pm 10 \).
- Testing \( x = 1 \):
\[
g(1) = 1^3 + 1^2 - 18(1) + 10 = 1 + 1 - 18 + 10 = -6 \quad (\text{not a root})
\]
- Testing \( x = -1 \):
\[
g(-1) = (-1)^3 + (-1)^2 - 18(-1) + 10 = -1 + 1 + 18 + 10 = 28 \quad (\text{not a root})
\]
- Testing \( x = 2 \):
\[
g(2) = 2^3 + 2^2 - 18(2) + 10 = 8 + 4 - 36 + 10 = -14 \quad (\text{not a root})
\]
- Testing \( x = 5 \):
\[
g(5) = 5^3 + 5^2 - 18(5) + 10 = 125 + 25 - 90 + 10 = 70 \quad (\text{not a root})
\]
- Testing \( x = -2 \):
\[
g(-2) = (-2)^3 + (-2)^2 - 18(-2) + 10 = -8 + 4 + 36 + 10 = 42 \quad (\text{not a root})
\]
- Testing \( x = -5 \):
\[
g(-5) = (-5)^3 + (-5)^2 - 18(-5) + 10 = -125 + 25 + 90 + 10 = -0 \quad (\text{is a root})
\]
### Step 5: Synthetic Division by \( x + 5 \)
\[
\begin{array}{r|rrrr}
-5 & 1 & 1 & -18 & 10 \\
& & -5 & 20 & -10 \\
\hline
& 1 & -4 & 2 & 0 \\
\end{array}
\]
Now we have \( g(x) = (x + 5)(x^2 - 4x + 2) \).
### Step 6: Solve \( x^2 - 4x + 2 = 0 \)
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 8}}{2} = \frac{4 \pm \sqrt{8}}{2} = \frac{4 \pm 2\sqrt{2}}{2} = 2 \pm \sqrt{2}.
\]
### Step 7: List All Zeros
The zeros of \( f(x) \) are:
\[
\boxed{2, -5, 2 + \sqrt{2}, 2 - \sqrt{2}}.
\]
### Step 1: Apply the Rational Root Theorem
The possible rational roots are the factors of the constant term (-20) divided by the factors of the leading coefficient (1). This gives us the possible rational roots:
\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20
\]
### Step 2: Test Possible Rational Roots
We'll test these candidates in the polynomial \( f(x) \).
- Testing \( x = 1 \):
\[
f(1) = 1^4 - 1^3 - 20(1^2) + 46(1) - 20 = 1 - 1 - 20 + 46 - 20 = 6 \quad (\text{not a root})
\]
- Testing \( x = 2 \):
\[
f(2) = 2^4 - 2^3 - 20(2^2) + 46(2) - 20 = 16 - 8 - 80 + 92 - 20 = 0 \quad (\text{is a root})
\]
Since \( x = 2 \) is a root, we can divide \( f(x) \) by \( x - 2 \).
### Step 3: Synthetic Division by \( x - 2 \)
\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & -20 & 46 & -20 \\
& & 2 & 2 & -36 & 20 \\
\hline
& 1 & 1 & -18 & 10 & 0 \\
\end{array}
\]
This gives us \( f(x) = (x - 2)(x^3 + x^2 - 18x + 10) \).
### Step 4: Find the Roots of \( x^3 + x^2 - 18x + 10 \)
We will apply the Rational Root Theorem again for \( g(x) = x^3 + x^2 - 18x + 10 \) with possible roots \( \pm 1, \pm 2, \pm 5, \pm 10 \).
- Testing \( x = 1 \):
\[
g(1) = 1^3 + 1^2 - 18(1) + 10 = 1 + 1 - 18 + 10 = -6 \quad (\text{not a root})
\]
- Testing \( x = -1 \):
\[
g(-1) = (-1)^3 + (-1)^2 - 18(-1) + 10 = -1 + 1 + 18 + 10 = 28 \quad (\text{not a root})
\]
- Testing \( x = 2 \):
\[
g(2) = 2^3 + 2^2 - 18(2) + 10 = 8 + 4 - 36 + 10 = -14 \quad (\text{not a root})
\]
- Testing \( x = 5 \):
\[
g(5) = 5^3 + 5^2 - 18(5) + 10 = 125 + 25 - 90 + 10 = 70 \quad (\text{not a root})
\]
- Testing \( x = -2 \):
\[
g(-2) = (-2)^3 + (-2)^2 - 18(-2) + 10 = -8 + 4 + 36 + 10 = 42 \quad (\text{not a root})
\]
- Testing \( x = -5 \):
\[
g(-5) = (-5)^3 + (-5)^2 - 18(-5) + 10 = -125 + 25 + 90 + 10 = -0 \quad (\text{is a root})
\]
### Step 5: Synthetic Division by \( x + 5 \)
\[
\begin{array}{r|rrrr}
-5 & 1 & 1 & -18 & 10 \\
& & -5 & 20 & -10 \\
\hline
& 1 & -4 & 2 & 0 \\
\end{array}
\]
Now we have \( g(x) = (x + 5)(x^2 - 4x + 2) \).
### Step 6: Solve \( x^2 - 4x + 2 = 0 \)
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 8}}{2} = \frac{4 \pm \sqrt{8}}{2} = \frac{4 \pm 2\sqrt{2}}{2} = 2 \pm \sqrt{2}.
\]
### Step 7: List All Zeros
The zeros of \( f(x) \) are:
\[
\boxed{2, -5, 2 + \sqrt{2}, 2 - \sqrt{2}}.
\]