Question
Use the Rational Zero Theorem to list all possible rational zeros for the given function.
f left parenthesis x right parenthesis equals 9 x Superscript 4 Baseline minus x cubed plus 5 x squared minus 3 x minus 6
Question content area bottom
Part 1
Which of the following is the complete list of possible zeros of the given function?
A.plus or minus 1 comma plus or minus 3 comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
plus or minus 1 comma plus or minus 3 comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
B.plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus one third comma plus or minus two thirds
plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus one third comma plus or minus two thirds
C.plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus 6 comma plus or minus one third comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus 6 comma plus or minus one third comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
D.
The function has no rational zeros
f left parenthesis x right parenthesis equals 9 x Superscript 4 Baseline minus x cubed plus 5 x squared minus 3 x minus 6
Question content area bottom
Part 1
Which of the following is the complete list of possible zeros of the given function?
A.plus or minus 1 comma plus or minus 3 comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
plus or minus 1 comma plus or minus 3 comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
B.plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus one third comma plus or minus two thirds
plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus one third comma plus or minus two thirds
C.plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus 6 comma plus or minus one third comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
plus or minus 1 comma plus or minus 3 comma plus or minus 2 comma plus or minus 6 comma plus or minus one third comma plus or minus two thirds comma plus or minus one ninth comma plus or minus two ninths
D.
The function has no rational zeros
Answers
Answered by
GPT-4o mini
To apply the Rational Zero Theorem, we first need to identify the coefficients of the polynomial \( f(x) = 9x^4 - x^3 + 5x^2 - 3x - 6 \). The coefficients are:
- Leading coefficient (a) = 9 (the coefficient of \( x^4 \))
- Constant term (c) = -6 (the constant term)
The Rational Zero Theorem states that any rational zero of the polynomial function can be expressed as \( \frac{p}{q} \) where:
- \( p \) is a factor of the constant term (\( c = -6 \))
- \( q \) is a factor of the leading coefficient (\( a = 9 \))
**Step 1: Find the factors of the constant term (-6)**
The factors of -6 are:
- ±1, ±2, ±3, ±6
**Step 2: Find the factors of the leading coefficient (9)**
The factors of 9 are:
- ±1, ±3, ±9
**Step 3: Form all possible rational zeros \( \frac{p}{q} \)**
We will divide each factor of -6 by each factor of 9:
1. Using \( p = 1 \):
- \( \frac{1}{1} = 1 \)
- \( \frac{1}{3} = \frac{1}{3} \)
- \( \frac{1}{9} = \frac{1}{9} \)
2. Using \( p = -1 \):
- \( \frac{-1}{1} = -1 \)
- \( \frac{-1}{3} = -\frac{1}{3} \)
- \( \frac{-1}{9} = -\frac{1}{9} \)
3. Using \( p = 2 \):
- \( \frac{2}{1} = 2 \)
- \( \frac{2}{3} = \frac{2}{3} \)
- \( \frac{2}{9} = \frac{2}{9} \)
4. Using \( p = -2 \):
- \( \frac{-2}{1} = -2 \)
- \( \frac{-2}{3} = -\frac{2}{3} \)
- \( \frac{-2}{9} = -\frac{2}{9} \)
5. Using \( p = 3 \):
- \( \frac{3}{1} = 3 \)
- \( \frac{3}{3} = 1 \) (already accounted for)
- \( \frac{3}{9} = \frac{1}{3} \) (already accounted for)
6. Using \( p = -3 \):
- \( \frac{-3}{1} = -3 \)
- \( \frac{-3}{3} = -1 \) (already accounted for)
- \( \frac{-3}{9} = -\frac{1}{3} \) (already accounted for)
7. Using \( p = 6 \):
- \( \frac{6}{1} = 6 \)
- \( \frac{6}{3} = 2 \) (already accounted for)
- \( \frac{6}{9} = \frac{2}{3} \) (already accounted for)
8. Using \( p = -6 \):
- \( \frac{-6}{1} = -6 \)
- \( \frac{-6}{3} = -2 \) (already accounted for)
- \( \frac{-6}{9} = -\frac{2}{3} \) (already accounted for)
**Step 4: List all unique rational zeros:**
From the above calculations, the list of possible rational zeros is:
- \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \)
Now, we can evaluate the provided options:
- **Option A**: \( \pm 1, \pm 3, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \) (missing \( \pm 2 \) and \( \pm 6 \))
- **Option B**: \( \pm 1, \pm 3, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3} \) (missing \( \pm 6 \) and the fractions \( \frac{1}{9}, \frac{2}{9} \))
- **Option C**: \( \pm 1, \pm 3, \pm 2, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \) (this option includes all the possible rational zeros found)
- **Option D**: The function has no rational zeros (not correct)
Thus, the complete list of possible rational zeros for the function is given by:
**C: \( \pm 1, \pm 3, \pm 2, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \)**.
- Leading coefficient (a) = 9 (the coefficient of \( x^4 \))
- Constant term (c) = -6 (the constant term)
The Rational Zero Theorem states that any rational zero of the polynomial function can be expressed as \( \frac{p}{q} \) where:
- \( p \) is a factor of the constant term (\( c = -6 \))
- \( q \) is a factor of the leading coefficient (\( a = 9 \))
**Step 1: Find the factors of the constant term (-6)**
The factors of -6 are:
- ±1, ±2, ±3, ±6
**Step 2: Find the factors of the leading coefficient (9)**
The factors of 9 are:
- ±1, ±3, ±9
**Step 3: Form all possible rational zeros \( \frac{p}{q} \)**
We will divide each factor of -6 by each factor of 9:
1. Using \( p = 1 \):
- \( \frac{1}{1} = 1 \)
- \( \frac{1}{3} = \frac{1}{3} \)
- \( \frac{1}{9} = \frac{1}{9} \)
2. Using \( p = -1 \):
- \( \frac{-1}{1} = -1 \)
- \( \frac{-1}{3} = -\frac{1}{3} \)
- \( \frac{-1}{9} = -\frac{1}{9} \)
3. Using \( p = 2 \):
- \( \frac{2}{1} = 2 \)
- \( \frac{2}{3} = \frac{2}{3} \)
- \( \frac{2}{9} = \frac{2}{9} \)
4. Using \( p = -2 \):
- \( \frac{-2}{1} = -2 \)
- \( \frac{-2}{3} = -\frac{2}{3} \)
- \( \frac{-2}{9} = -\frac{2}{9} \)
5. Using \( p = 3 \):
- \( \frac{3}{1} = 3 \)
- \( \frac{3}{3} = 1 \) (already accounted for)
- \( \frac{3}{9} = \frac{1}{3} \) (already accounted for)
6. Using \( p = -3 \):
- \( \frac{-3}{1} = -3 \)
- \( \frac{-3}{3} = -1 \) (already accounted for)
- \( \frac{-3}{9} = -\frac{1}{3} \) (already accounted for)
7. Using \( p = 6 \):
- \( \frac{6}{1} = 6 \)
- \( \frac{6}{3} = 2 \) (already accounted for)
- \( \frac{6}{9} = \frac{2}{3} \) (already accounted for)
8. Using \( p = -6 \):
- \( \frac{-6}{1} = -6 \)
- \( \frac{-6}{3} = -2 \) (already accounted for)
- \( \frac{-6}{9} = -\frac{2}{3} \) (already accounted for)
**Step 4: List all unique rational zeros:**
From the above calculations, the list of possible rational zeros is:
- \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \)
Now, we can evaluate the provided options:
- **Option A**: \( \pm 1, \pm 3, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \) (missing \( \pm 2 \) and \( \pm 6 \))
- **Option B**: \( \pm 1, \pm 3, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3} \) (missing \( \pm 6 \) and the fractions \( \frac{1}{9}, \frac{2}{9} \))
- **Option C**: \( \pm 1, \pm 3, \pm 2, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \) (this option includes all the possible rational zeros found)
- **Option D**: The function has no rational zeros (not correct)
Thus, the complete list of possible rational zeros for the function is given by:
**C: \( \pm 1, \pm 3, \pm 2, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \)**.