Question
Find all zeros of the polynomial function. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root.
f left parenthesis x right parenthesis equals 2 x Superscript 4 Baseline minus 45 x cubed plus 215 x squared plus 405 x plus 143
Question content area bottom
Part 1
The zeros of the function are
enter your response here.
(Type integers or simplified fractions. Use a comma to separate answers as needed.)
f left parenthesis x right parenthesis equals 2 x Superscript 4 Baseline minus 45 x cubed plus 215 x squared plus 405 x plus 143
Question content area bottom
Part 1
The zeros of the function are
enter your response here.
(Type integers or simplified fractions. Use a comma to separate answers as needed.)
Answers
GPT-4o mini
To find the zeros of the polynomial function \( f(x) = 2x^4 - 45x^3 + 215x^2 + 405x + 143 \), we can use the Rational Root Theorem, Descartes' Rule of Signs, and potentially graph the function for more insight.
### Step 1: Rational Root Theorem
The Rational Root Theorem states that any rational root of the polynomial of the form \( \frac{p}{q} \) implies that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
- The constant term is \( 143 \).
- The factors of \( 143 \) are \( \pm 1, \pm 11, \pm 13, \pm 143 \).
- The leading coefficient is \( 2 \).
- The factors of \( 2 \) are \( \pm 1, \pm 2 \).
Thus, the possible rational roots are:
\[
\pm 1, \pm \frac{1}{2}, \pm 11, \pm \frac{11}{2}, \pm 13, \pm \frac{13}{2}, \pm 143, \pm \frac{143}{2}
\]
### Step 2: Descartes' Rule of Signs
Next, we will apply Descartes' Rule of Signs to determine the number of positive and negative real roots.
1. For positive roots, look at \( f(x) \):
- Changes of sign in \( 2x^4 - 45x^3 + 215x^2 + 405x + 143 \):
- \( 2 \) to \(-45\) (1 change)
- \(-45\) to \(215\) (1 change)
- \(215\) to \(405\) (no change)
- \(405\) to \(143\) (no change)
- Total sign changes = 2, leading to possible 2 or 0 positive real roots.
2. For negative roots, look at \( f(-x) \):
\[
f(-x) = 2(-x)^4 - 45(-x)^3 + 215(-x)^2 + 405(-x) + 143 = 2x^4 + 45x^3 + 215x^2 - 405x + 143
\]
- Changes of sign:
- \( 2 \) to \(45\) (no change)
- \(45\) to \(215\) (no change)
- \(215\) to \(-405\) (1 change)
- \(-405\) to \(143\) (1 change)
- Total sign changes = 2, leading to possible 2 or 0 negative real roots.
### Step 3: Graphing the Function
To locate the first root, use a graphing utility to visualize where the function crosses the x-axis.
### Step 4: Testing Possible Rational Roots
Let's test some potential rational roots:
1. **Test \( x = 1 \)**:
\[
f(1) = 2(1)^4 - 45(1)^3 + 215(1)^2 + 405(1) + 143 = 2 - 45 + 215 + 405 + 143 = 720 \, (\text{not a root})
\]
2. **Test \( x = -1 \)**:
\[
f(-1) = 2(-1)^4 - 45(-1)^3 + 215(-1)^2 + 405(-1) + 143 = 2 + 45 + 215 - 405 + 143 = 0 \, (\text{a root})
\]
### Step 5: Polynomial Division
Since \( x = -1 \) is a root, we can perform synthetic or polynomial long division on \( f(x) \) by \( x + 1 \):
Perform synthetic division:
\[
\begin{array}{r|rrrrr}
-1 & 2 & -45 & 215 & 405 & 143 \\
& & -2 & 47 & -262 & -143 \\
\hline
& 2 & -47 & 262 & 143 & 0 \\
\end{array}
\]
This gives the quotient \( 2x^3 - 47x^2 + 262x + 143 \).
### Step 6: Finding Remaining Zeros
Now we need to find the zeros of \( 2x^3 - 47x^2 + 262x + 143 \). We can use the Rational Root Theorem again and check for roots. Testing \( x = 11 \):
\[
f(11) = 2(11)^3 - 47(11)^2 + 262(11) + 143 = 2(1331) - 47(121) + 2882 + 143
= 2662 - 5687 + 2882 + 143 = 0 \, (\text{a root})
\]
We can then use synthetic division again to divide \( 2x^3 - 47x^2 + 262x + 143 \) by \( x - 11 \):
1. Perform synthetic division with \( x = 11 \):
\[
\begin{array}{r|rrrr}
11 & 2 & -47 & 262 & 143 \\
& & 22 & -275 & -143 \\
\hline
& 2 & -25 & -13 & 0 \\
\end{array}
\]
This gives the quotient \( 2x^2 - 25x - 13 \).
### Step 7: Solving the Quadratic
Now solve \( 2x^2 - 25x - 13 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 2 \cdot (-13)}}{2 \cdot 2}
= \frac{25 \pm \sqrt{625 + 104}}{4} = \frac{25 \pm \sqrt{729}}{4} = \frac{25 \pm 27}{4}
\]
This gives:
\[
x_1 = \frac{52}{4} = 13, \quad x_2 = \frac{-2}{4} = -\frac{1}{2}
\]
### Final Zeros
Thus, the zeros of the polynomial function are:
\[
\boxed{-1, 11, 13, -\frac{1}{2}}
\]
### Step 1: Rational Root Theorem
The Rational Root Theorem states that any rational root of the polynomial of the form \( \frac{p}{q} \) implies that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
- The constant term is \( 143 \).
- The factors of \( 143 \) are \( \pm 1, \pm 11, \pm 13, \pm 143 \).
- The leading coefficient is \( 2 \).
- The factors of \( 2 \) are \( \pm 1, \pm 2 \).
Thus, the possible rational roots are:
\[
\pm 1, \pm \frac{1}{2}, \pm 11, \pm \frac{11}{2}, \pm 13, \pm \frac{13}{2}, \pm 143, \pm \frac{143}{2}
\]
### Step 2: Descartes' Rule of Signs
Next, we will apply Descartes' Rule of Signs to determine the number of positive and negative real roots.
1. For positive roots, look at \( f(x) \):
- Changes of sign in \( 2x^4 - 45x^3 + 215x^2 + 405x + 143 \):
- \( 2 \) to \(-45\) (1 change)
- \(-45\) to \(215\) (1 change)
- \(215\) to \(405\) (no change)
- \(405\) to \(143\) (no change)
- Total sign changes = 2, leading to possible 2 or 0 positive real roots.
2. For negative roots, look at \( f(-x) \):
\[
f(-x) = 2(-x)^4 - 45(-x)^3 + 215(-x)^2 + 405(-x) + 143 = 2x^4 + 45x^3 + 215x^2 - 405x + 143
\]
- Changes of sign:
- \( 2 \) to \(45\) (no change)
- \(45\) to \(215\) (no change)
- \(215\) to \(-405\) (1 change)
- \(-405\) to \(143\) (1 change)
- Total sign changes = 2, leading to possible 2 or 0 negative real roots.
### Step 3: Graphing the Function
To locate the first root, use a graphing utility to visualize where the function crosses the x-axis.
### Step 4: Testing Possible Rational Roots
Let's test some potential rational roots:
1. **Test \( x = 1 \)**:
\[
f(1) = 2(1)^4 - 45(1)^3 + 215(1)^2 + 405(1) + 143 = 2 - 45 + 215 + 405 + 143 = 720 \, (\text{not a root})
\]
2. **Test \( x = -1 \)**:
\[
f(-1) = 2(-1)^4 - 45(-1)^3 + 215(-1)^2 + 405(-1) + 143 = 2 + 45 + 215 - 405 + 143 = 0 \, (\text{a root})
\]
### Step 5: Polynomial Division
Since \( x = -1 \) is a root, we can perform synthetic or polynomial long division on \( f(x) \) by \( x + 1 \):
Perform synthetic division:
\[
\begin{array}{r|rrrrr}
-1 & 2 & -45 & 215 & 405 & 143 \\
& & -2 & 47 & -262 & -143 \\
\hline
& 2 & -47 & 262 & 143 & 0 \\
\end{array}
\]
This gives the quotient \( 2x^3 - 47x^2 + 262x + 143 \).
### Step 6: Finding Remaining Zeros
Now we need to find the zeros of \( 2x^3 - 47x^2 + 262x + 143 \). We can use the Rational Root Theorem again and check for roots. Testing \( x = 11 \):
\[
f(11) = 2(11)^3 - 47(11)^2 + 262(11) + 143 = 2(1331) - 47(121) + 2882 + 143
= 2662 - 5687 + 2882 + 143 = 0 \, (\text{a root})
\]
We can then use synthetic division again to divide \( 2x^3 - 47x^2 + 262x + 143 \) by \( x - 11 \):
1. Perform synthetic division with \( x = 11 \):
\[
\begin{array}{r|rrrr}
11 & 2 & -47 & 262 & 143 \\
& & 22 & -275 & -143 \\
\hline
& 2 & -25 & -13 & 0 \\
\end{array}
\]
This gives the quotient \( 2x^2 - 25x - 13 \).
### Step 7: Solving the Quadratic
Now solve \( 2x^2 - 25x - 13 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 2 \cdot (-13)}}{2 \cdot 2}
= \frac{25 \pm \sqrt{625 + 104}}{4} = \frac{25 \pm \sqrt{729}}{4} = \frac{25 \pm 27}{4}
\]
This gives:
\[
x_1 = \frac{52}{4} = 13, \quad x_2 = \frac{-2}{4} = -\frac{1}{2}
\]
### Final Zeros
Thus, the zeros of the polynomial function are:
\[
\boxed{-1, 11, 13, -\frac{1}{2}}
\]