Answer :
Sure! Let's use the Rational Zero Theorem to list all possible rational zeros of the polynomial [tex]\( f(x) = 9x^4 - x^3 + 3x^2 - 4x - 21 \)[/tex].
### Step 1: Identify the Constant Term and the Leading Coefficient
1. The constant term of the polynomial [tex]\( f(x) = 9x^4 - x^3 + 3x^2 - 4x - 21 \)[/tex] is [tex]\(-21\)[/tex].
2. The leading coefficient (the coefficient of the highest degree term, [tex]\( x^4 \)[/tex]) is [tex]\( 9 \)[/tex].
### Step 2: Find the Factors of the Constant Term and the Leading Coefficient
#### Factors of the Constant Term ([tex]\(-21\)[/tex])
We find all factors of [tex]\(-21\)[/tex]. These include both positive and negative factors:
[tex]\[ \pm 1, \pm 3, \pm 7, \pm 21 \][/tex]
#### Factors of the Leading Coefficient ([tex]\(9\)[/tex])
We find all factors of [tex]\(9\)[/tex]. These include both positive and negative factors:
[tex]\[ \pm 1, \pm 3, \pm 9 \][/tex]
### Step 3: Form the Ratios of All Factors of the Constant Term Over Factors of the Leading Coefficient
According to the Rational Zero Theorem, all possible rational zeros of the polynomial [tex]\( f(x) \)[/tex] are of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
Let's list all possible combinations:
#### Possible Ratios ([tex]\( p \)[/tex] over [tex]\( q \)[/tex]):
1. [tex]\(\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 3}, \frac{\pm 1}{\pm 9}\)[/tex]
2. [tex]\(\frac{\pm 3}{\pm 1}, \frac{\pm 3}{\pm 3}, \frac{\pm 3}{\pm 9}\)[/tex]
3. [tex]\(\frac{\pm 7}{\pm 1}, \frac{\pm 7}{\pm 3}, \frac{\pm 7}{\pm 9}\)[/tex]
4. [tex]\(\frac{\pm 21}{\pm 1}, \frac{\pm 21}{\pm 3}, \frac{\pm 21}{\pm 9}\)[/tex]
Simplifying these fractions, we get the following list of possible rational zeros:
#### Simplified Possible Rational Zeros:
1. [tex]\(\pm 1, \pm 3, \pm 7, \pm 21\)[/tex]
2. [tex]\(\pm \frac{1}{3}, \pm 1, \pm \frac{1}{9}\)[/tex]
3. [tex]\(\pm 3, \pm 1, \pm \frac{3}{9} = \pm \frac{1}{3}\)[/tex]
4. [tex]\(\pm \frac{7}{3}, \pm \frac{7}{9}\)[/tex]
5. [tex]\(\pm 21, \pm 7, \pm 3, \pm 1, \pm \frac{21}{9} = \pm \frac{7}{3}\)[/tex]
Combining and removing any duplicates, we have the following possible rational zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{7}{3}, \pm \frac{7}{9} \right\} \][/tex]
### Final List of Possible Rational Zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{7}{3}, \pm \frac{7}{9} \right\} \][/tex]
Converted to decimal form (and sorted for clarity):
[tex]\[ \left\{ 1.0, 3.0, 7.0, 21.0, \frac{1}{3} \approx 0.333, \frac{1}{9} \approx 0.111, \frac{7}{3} \approx 2.333, \frac{7}{9} \approx 0.778, -1.0, -3.0, -7.0, -21.0, -\frac{1}{3} \approx -0.333, -\frac{1}{9} \approx -0.111, -\frac{7}{3} \approx -2.333, -\frac{7}{9} \approx -0.778 \right\} \][/tex]
### Complete List of Possible Rational Zeros:
[tex]\[ \left\{ 0.333, 1.0, 0.111, -0.333, -0.111, 3.0, 2.333, 7.0, 0.778, -0.778, -21.0, 21.0, -2.333, -7.0, -3.0, -1.0 \right\} \][/tex]
These are all the possible rational zeros for the polynomial [tex]\( f(x) = 9x^4 - x^3 + 3x^2 - 4x - 21 \)[/tex].
### Step 1: Identify the Constant Term and the Leading Coefficient
1. The constant term of the polynomial [tex]\( f(x) = 9x^4 - x^3 + 3x^2 - 4x - 21 \)[/tex] is [tex]\(-21\)[/tex].
2. The leading coefficient (the coefficient of the highest degree term, [tex]\( x^4 \)[/tex]) is [tex]\( 9 \)[/tex].
### Step 2: Find the Factors of the Constant Term and the Leading Coefficient
#### Factors of the Constant Term ([tex]\(-21\)[/tex])
We find all factors of [tex]\(-21\)[/tex]. These include both positive and negative factors:
[tex]\[ \pm 1, \pm 3, \pm 7, \pm 21 \][/tex]
#### Factors of the Leading Coefficient ([tex]\(9\)[/tex])
We find all factors of [tex]\(9\)[/tex]. These include both positive and negative factors:
[tex]\[ \pm 1, \pm 3, \pm 9 \][/tex]
### Step 3: Form the Ratios of All Factors of the Constant Term Over Factors of the Leading Coefficient
According to the Rational Zero Theorem, all possible rational zeros of the polynomial [tex]\( f(x) \)[/tex] are of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
Let's list all possible combinations:
#### Possible Ratios ([tex]\( p \)[/tex] over [tex]\( q \)[/tex]):
1. [tex]\(\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 3}, \frac{\pm 1}{\pm 9}\)[/tex]
2. [tex]\(\frac{\pm 3}{\pm 1}, \frac{\pm 3}{\pm 3}, \frac{\pm 3}{\pm 9}\)[/tex]
3. [tex]\(\frac{\pm 7}{\pm 1}, \frac{\pm 7}{\pm 3}, \frac{\pm 7}{\pm 9}\)[/tex]
4. [tex]\(\frac{\pm 21}{\pm 1}, \frac{\pm 21}{\pm 3}, \frac{\pm 21}{\pm 9}\)[/tex]
Simplifying these fractions, we get the following list of possible rational zeros:
#### Simplified Possible Rational Zeros:
1. [tex]\(\pm 1, \pm 3, \pm 7, \pm 21\)[/tex]
2. [tex]\(\pm \frac{1}{3}, \pm 1, \pm \frac{1}{9}\)[/tex]
3. [tex]\(\pm 3, \pm 1, \pm \frac{3}{9} = \pm \frac{1}{3}\)[/tex]
4. [tex]\(\pm \frac{7}{3}, \pm \frac{7}{9}\)[/tex]
5. [tex]\(\pm 21, \pm 7, \pm 3, \pm 1, \pm \frac{21}{9} = \pm \frac{7}{3}\)[/tex]
Combining and removing any duplicates, we have the following possible rational zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{7}{3}, \pm \frac{7}{9} \right\} \][/tex]
### Final List of Possible Rational Zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm 7, \pm 21, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{7}{3}, \pm \frac{7}{9} \right\} \][/tex]
Converted to decimal form (and sorted for clarity):
[tex]\[ \left\{ 1.0, 3.0, 7.0, 21.0, \frac{1}{3} \approx 0.333, \frac{1}{9} \approx 0.111, \frac{7}{3} \approx 2.333, \frac{7}{9} \approx 0.778, -1.0, -3.0, -7.0, -21.0, -\frac{1}{3} \approx -0.333, -\frac{1}{9} \approx -0.111, -\frac{7}{3} \approx -2.333, -\frac{7}{9} \approx -0.778 \right\} \][/tex]
### Complete List of Possible Rational Zeros:
[tex]\[ \left\{ 0.333, 1.0, 0.111, -0.333, -0.111, 3.0, 2.333, 7.0, 0.778, -0.778, -21.0, 21.0, -2.333, -7.0, -3.0, -1.0 \right\} \][/tex]
These are all the possible rational zeros for the polynomial [tex]\( f(x) = 9x^4 - x^3 + 3x^2 - 4x - 21 \)[/tex].
