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To identify all potential rational roots of the polynomial [tex]\( f(x) = 3x^2 - x - 4 \)[/tex], we can use the Rational Root Theorem. According to this theorem, any potential rational root of a polynomial is in the form [tex]\( \frac{p}{q} \)[/tex], where:
- [tex]\( p \)[/tex] is a factor of the constant term (the term without [tex]\( x \)[/tex]).
- [tex]\( q \)[/tex] is a factor of the leading coefficient (the coefficient of the highest degree term).
Let's go through each step in detail:
1. Identify the constant term and the leading coefficient:
- The constant term in the polynomial [tex]\( f(x) = 3x^2 - x - 4 \)[/tex] is [tex]\(-4\)[/tex].
- The leading coefficient (the coefficient of [tex]\( x^2 \)[/tex]) is [tex]\(3\)[/tex].
2. Find the factors of the constant term [tex]\(-4\)[/tex]:
- The factors of [tex]\(-4\)[/tex] are [tex]\( \pm 1, \pm 2, \pm 4 \)[/tex].
3. Find the factors of the leading coefficient [tex]\(3\)[/tex]:
- The factors of [tex]\(3\)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
So, we can fill in the blanks as follows:
- Values of [tex]\( p \)[/tex] are factors of [tex]\(\mathbf{-4}\)[/tex].
- Values of [tex]\( q \)[/tex] are factors of [tex]\(\mathbf{3}\)[/tex].
Now, let's list all potential rational roots by forming the fractions [tex]\( \frac{p}{q} \)[/tex] for each combination of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- Possible values of [tex]\( p \)[/tex]: [tex]\( -4, -2, -1, 1, 2, 4 \)[/tex].
- Possible values of [tex]\( q \)[/tex]: [tex]\( -3, -1, 1, 3 \)[/tex].
Forming every possible fraction [tex]\( \frac{p}{q} \)[/tex] and considering both positive and negative factors:
- For [tex]\( p = -4 \)[/tex]: potential roots are [tex]\( \frac{-4}{-3}, \frac{-4}{-1}, \frac{-4}{1}, \frac{-4}{3} \)[/tex] which simplifies to [tex]\( \frac{4}{3}, 4, -4, -\frac{4}{3} \)[/tex].
- For [tex]\( p = -2 \)[/tex]: potential roots are [tex]\( \frac{-2}{-3}, \frac{-2}{-1}, \frac{-2}{1}, \frac{-2}{3} \)[/tex] which simplifies to [tex]\( \frac{2}{3}, 2, -2, -\frac{2}{3} \)[/tex].
- For [tex]\( p = -1 \)[/tex]: potential roots are [tex]\( \frac{-1}{-3}, \frac{-1}{-1}, \frac{-1}{1}, \frac{-1}{3} \)[/tex] which simplifies to [tex]\( \frac{1}{3}, 1, -1, -\frac{1}{3} \)[/tex].
- For [tex]\( p = 1 \)[/tex]: potential roots are [tex]\( \frac{1}{-3}, \frac{1}{-1}, \frac{1}{1}, \frac{1}{3} \)[/tex] which simplifies to [tex]\( -\frac{1}{3}, -1, 1, \frac{1}{3} \)[/tex].
- For [tex]\( p = 2 \)[/tex]: potential roots are [tex]\( \frac{2}{-3}, \frac{2}{-1}, \frac{2}{1}, \frac{2}{3} \)[/tex] which simplifies to [tex]\( -\frac{2}{3}, -2, 2, \frac{2}{3} \)[/tex].
- For [tex]\( p = 4 \)[/tex]: potential roots are [tex]\( \frac{4}{-3}, \frac{4}{-1}, \frac{4}{1}, \frac{4}{3} \)[/tex] which simplifies to [tex]\( -\frac{4}{3}, -4, 4, \frac{4}{3} \)[/tex].
Combining and removing duplicates, we get the list of potential rational roots:
[tex]\[ \left\{ -4.0, -2.0, -1.3333333333333333, -1.0, -0.6666666666666666, -0.3333333333333333, 0.3333333333333333, 0.6666666666666666, 1.0, 1.3333333333333333, 2.0, 4.0 \right\}. \][/tex]
Thus, the potential rational roots of [tex]\( f(x) = 3x^2 - x - 4 \)[/tex] are:
[tex]\[ -4, -2, -1.333, -1, -0.667, -0.333, 0.333, 0.667, 1, 1.333, 2, 4. \][/tex]
- [tex]\( p \)[/tex] is a factor of the constant term (the term without [tex]\( x \)[/tex]).
- [tex]\( q \)[/tex] is a factor of the leading coefficient (the coefficient of the highest degree term).
Let's go through each step in detail:
1. Identify the constant term and the leading coefficient:
- The constant term in the polynomial [tex]\( f(x) = 3x^2 - x - 4 \)[/tex] is [tex]\(-4\)[/tex].
- The leading coefficient (the coefficient of [tex]\( x^2 \)[/tex]) is [tex]\(3\)[/tex].
2. Find the factors of the constant term [tex]\(-4\)[/tex]:
- The factors of [tex]\(-4\)[/tex] are [tex]\( \pm 1, \pm 2, \pm 4 \)[/tex].
3. Find the factors of the leading coefficient [tex]\(3\)[/tex]:
- The factors of [tex]\(3\)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
So, we can fill in the blanks as follows:
- Values of [tex]\( p \)[/tex] are factors of [tex]\(\mathbf{-4}\)[/tex].
- Values of [tex]\( q \)[/tex] are factors of [tex]\(\mathbf{3}\)[/tex].
Now, let's list all potential rational roots by forming the fractions [tex]\( \frac{p}{q} \)[/tex] for each combination of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- Possible values of [tex]\( p \)[/tex]: [tex]\( -4, -2, -1, 1, 2, 4 \)[/tex].
- Possible values of [tex]\( q \)[/tex]: [tex]\( -3, -1, 1, 3 \)[/tex].
Forming every possible fraction [tex]\( \frac{p}{q} \)[/tex] and considering both positive and negative factors:
- For [tex]\( p = -4 \)[/tex]: potential roots are [tex]\( \frac{-4}{-3}, \frac{-4}{-1}, \frac{-4}{1}, \frac{-4}{3} \)[/tex] which simplifies to [tex]\( \frac{4}{3}, 4, -4, -\frac{4}{3} \)[/tex].
- For [tex]\( p = -2 \)[/tex]: potential roots are [tex]\( \frac{-2}{-3}, \frac{-2}{-1}, \frac{-2}{1}, \frac{-2}{3} \)[/tex] which simplifies to [tex]\( \frac{2}{3}, 2, -2, -\frac{2}{3} \)[/tex].
- For [tex]\( p = -1 \)[/tex]: potential roots are [tex]\( \frac{-1}{-3}, \frac{-1}{-1}, \frac{-1}{1}, \frac{-1}{3} \)[/tex] which simplifies to [tex]\( \frac{1}{3}, 1, -1, -\frac{1}{3} \)[/tex].
- For [tex]\( p = 1 \)[/tex]: potential roots are [tex]\( \frac{1}{-3}, \frac{1}{-1}, \frac{1}{1}, \frac{1}{3} \)[/tex] which simplifies to [tex]\( -\frac{1}{3}, -1, 1, \frac{1}{3} \)[/tex].
- For [tex]\( p = 2 \)[/tex]: potential roots are [tex]\( \frac{2}{-3}, \frac{2}{-1}, \frac{2}{1}, \frac{2}{3} \)[/tex] which simplifies to [tex]\( -\frac{2}{3}, -2, 2, \frac{2}{3} \)[/tex].
- For [tex]\( p = 4 \)[/tex]: potential roots are [tex]\( \frac{4}{-3}, \frac{4}{-1}, \frac{4}{1}, \frac{4}{3} \)[/tex] which simplifies to [tex]\( -\frac{4}{3}, -4, 4, \frac{4}{3} \)[/tex].
Combining and removing duplicates, we get the list of potential rational roots:
[tex]\[ \left\{ -4.0, -2.0, -1.3333333333333333, -1.0, -0.6666666666666666, -0.3333333333333333, 0.3333333333333333, 0.6666666666666666, 1.0, 1.3333333333333333, 2.0, 4.0 \right\}. \][/tex]
Thus, the potential rational roots of [tex]\( f(x) = 3x^2 - x - 4 \)[/tex] are:
[tex]\[ -4, -2, -1.333, -1, -0.667, -0.333, 0.333, 0.667, 1, 1.333, 2, 4. \][/tex]
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