Answer :
Sure! Let's apply the Rational Zero Theorem to the polynomial [tex]\( f(x) = 9x^4 - x^3 + 5x^2 - 2x - 6 \)[/tex] to list all possible rational zeros. The Rational Zero Theorem states that any rational zero, [tex]\( \frac{p}{q} \)[/tex], of a polynomial with integer coefficients, is such that [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
Here’s the step-by-step process:
1. Identify the constant term and the leading coefficient:
- The constant term [tex]\( a_0 \)[/tex] is [tex]\(-6\)[/tex].
- The leading coefficient [tex]\( a_n \)[/tex] (coefficient of the highest degree term) is [tex]\(9\)[/tex].
2. List all factors of the constant term ([tex]\( a_0 = -6 \)[/tex]):
Factors of [tex]\(-6\)[/tex] include: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
3. List all factors of the leading coefficient ([tex]\( a_n = 9 \)[/tex]):
Factors of [tex]\(9\)[/tex] include: [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
4. Form all possible rational zeros [tex]\( \frac{p}{q} \)[/tex]:
For each factor [tex]\( p \)[/tex] of the constant term and each factor [tex]\( q \)[/tex] of the leading coefficient, we form the fractions [tex]\( \frac{p}{q} \)[/tex] and [tex]\( \frac{-p}{q} \)[/tex].
- Possible factors [tex]\( p \)[/tex] of the constant term [tex]\(-6\)[/tex]: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
- Possible factors [tex]\( q \)[/tex] of the leading coefficient [tex]\(9\)[/tex]: [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
5. Generate all combinations of [tex]\( \frac{p}{q} \)[/tex] and simplify:
- When [tex]\( q = 1 \)[/tex]: [tex]\( \frac{\pm 1}{1} = \pm 1 \)[/tex], [tex]\( \frac{\pm 2}{1} = \pm 2 \)[/tex], [tex]\( \frac{\pm 3}{1} = \pm 3 \)[/tex], [tex]\( \frac{\pm 6}{1} = \pm 6 \)[/tex].
- When [tex]\( q = 3 \)[/tex]: [tex]\( \frac{\pm 1}{3} = \pm \frac{1}{3} \)[/tex], [tex]\( \frac{\pm 2}{3} = \pm \frac{2}{3} \)[/tex], [tex]\( \frac{\pm 3}{3} = \pm 1 \)[/tex], [tex]\( \frac{\pm 6}{3} = \pm 2 \)[/tex].
- When [tex]\( q = 9 \)[/tex]: [tex]\( \frac{\pm 1}{9} = \pm \frac{1}{9} \)[/tex], [tex]\( \frac{\pm 2}{9} = \pm \frac{2}{9} \)[/tex], [tex]\( \frac{\pm 3}{9} = \pm \frac{1}{3} \)[/tex], [tex]\( \frac{\pm 6}{9} = \pm \frac{2}{3} \)[/tex].
6. Combine and remove duplicates:
Listing all unique combinations from above, we have the following possible rational zeros:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \][/tex]
So, the list of all possible rational zeros of the polynomial [tex]\( f(x) = 9x^4 - x^3 + 5x^2 - 2x - 6 \)[/tex] is:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \][/tex]
Here’s the step-by-step process:
1. Identify the constant term and the leading coefficient:
- The constant term [tex]\( a_0 \)[/tex] is [tex]\(-6\)[/tex].
- The leading coefficient [tex]\( a_n \)[/tex] (coefficient of the highest degree term) is [tex]\(9\)[/tex].
2. List all factors of the constant term ([tex]\( a_0 = -6 \)[/tex]):
Factors of [tex]\(-6\)[/tex] include: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
3. List all factors of the leading coefficient ([tex]\( a_n = 9 \)[/tex]):
Factors of [tex]\(9\)[/tex] include: [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
4. Form all possible rational zeros [tex]\( \frac{p}{q} \)[/tex]:
For each factor [tex]\( p \)[/tex] of the constant term and each factor [tex]\( q \)[/tex] of the leading coefficient, we form the fractions [tex]\( \frac{p}{q} \)[/tex] and [tex]\( \frac{-p}{q} \)[/tex].
- Possible factors [tex]\( p \)[/tex] of the constant term [tex]\(-6\)[/tex]: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
- Possible factors [tex]\( q \)[/tex] of the leading coefficient [tex]\(9\)[/tex]: [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
5. Generate all combinations of [tex]\( \frac{p}{q} \)[/tex] and simplify:
- When [tex]\( q = 1 \)[/tex]: [tex]\( \frac{\pm 1}{1} = \pm 1 \)[/tex], [tex]\( \frac{\pm 2}{1} = \pm 2 \)[/tex], [tex]\( \frac{\pm 3}{1} = \pm 3 \)[/tex], [tex]\( \frac{\pm 6}{1} = \pm 6 \)[/tex].
- When [tex]\( q = 3 \)[/tex]: [tex]\( \frac{\pm 1}{3} = \pm \frac{1}{3} \)[/tex], [tex]\( \frac{\pm 2}{3} = \pm \frac{2}{3} \)[/tex], [tex]\( \frac{\pm 3}{3} = \pm 1 \)[/tex], [tex]\( \frac{\pm 6}{3} = \pm 2 \)[/tex].
- When [tex]\( q = 9 \)[/tex]: [tex]\( \frac{\pm 1}{9} = \pm \frac{1}{9} \)[/tex], [tex]\( \frac{\pm 2}{9} = \pm \frac{2}{9} \)[/tex], [tex]\( \frac{\pm 3}{9} = \pm \frac{1}{3} \)[/tex], [tex]\( \frac{\pm 6}{9} = \pm \frac{2}{3} \)[/tex].
6. Combine and remove duplicates:
Listing all unique combinations from above, we have the following possible rational zeros:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \][/tex]
So, the list of all possible rational zeros of the polynomial [tex]\( f(x) = 9x^4 - x^3 + 5x^2 - 2x - 6 \)[/tex] is:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{9}, \pm \frac{2}{9} \][/tex]
