Math / Algebra

QuestionUse the rational zeros theorem to list all possible rational zeros of the following. $f(x)=-25 x^{3}-2 x^{2}+4 x-1$
Be sure that no value in your list appears more than once.

Studdy Solution

STEP 1

1. The Rational Zeros Theorem states that any rational zero of a polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ is of the form $\frac{p}{q}$ , where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$ .
2. We need to identify the factors of the constant term and the leading coefficient.

STEP 2

1. Identify the constant term and the leading coefficient.
2. List all factors of the constant term.
3. List all factors of the leading coefficient.
4. Use the Rational Zeros Theorem to list all possible rational zeros.
5. Ensure no duplicate values in the list.

STEP 3

Identify the constant term and the leading coefficient of the polynomial $f(x) = -25x^3 - 2x^2 + 4x - 1$ .
The constant term $a_0$ is $-1$ .
The leading coefficient $a_n$ is $-25$ .

STEP 4

List all factors of the constant term $-1$ .
The factors of $-1$ are $\pm 1$ .

STEP 5

List all factors of the leading coefficient $-25$ .
The factors of $-25$ are $\pm 1, \pm 5, \pm 25$ .

STEP 6

Use the Rational Zeros Theorem to list all possible rational zeros, which are of the form $\frac{p}{q}$ , where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
Possible rational zeros are:
$\pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{1}{25}$
Simplifying, we get:
$\pm 1, \pm \frac{1}{5}, \pm \frac{1}{25}$

STEP 7

Ensure no duplicate values in the list.
The list of possible rational zeros is:
$\boxed{\pm 1, \pm \frac{1}{5}, \pm \frac{1}{25}}$

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QuestionUse the rational zeros theorem to list all possible rational zeros of the following. $f(x)=-25 x^{3}-2 x^{2}+4 x-1$
Be sure that no value in your list appears more than once.

Studdy Solution

STEP 1

STEP 2

1. Identify the constant term and the leading coefficient.
2. List all factors of the constant term.
3. List all factors of the leading coefficient.
4. Use the Rational Zeros Theorem to list all possible rational zeros.
5. Ensure no duplicate values in the list.

STEP 3

Identify the constant term and the leading coefficient of the polynomial $f(x) = -25x^3 - 2x^2 + 4x - 1$ .
The constant term $a_0$ is $-1$ .
The leading coefficient $a_n$ is $-25$ .

STEP 4

List all factors of the constant term $-1$ .
The factors of $-1$ are $\pm 1$ .

STEP 5

List all factors of the leading coefficient $-25$ .
The factors of $-25$ are $\pm 1, \pm 5, \pm 25$ .

STEP 6

STEP 7

Ensure no duplicate values in the list.
The list of possible rational zeros is:
$\boxed{\pm 1, \pm \frac{1}{5}, \pm \frac{1}{25}}$

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QuestionUse the rational zeros theorem to list all possible rational zeros of the following. f(x)=−25x3−2x2+4x−1f(x)=-25 x^{3}-2 x^{2}+4 x-1f(x)=−25x3−2x2+4x−1Be sure that no value in your list appears more than once.

Studdy Solution

STEP 1

STEP 2

1. Identify the constant term and the leading coefficient.2. List all factors of the constant term.3. List all factors of the leading coefficient.4. Use the Rational Zeros Theorem to list all possible rational zeros.5. Ensure no duplicate values in the list.

STEP 3

Identify the constant term and the leading coefficient of the polynomial f(x)=−25x3−2x2+4x−1 f(x) = -25x^3 - 2x^2 + 4x - 1 f(x)=−25x3−2x2+4x−1.The constant term a0 a_0 a0​ is −1-1−1.The leading coefficient an a_n an​ is −25-25−25.

STEP 4

List all factors of the constant term −1-1−1.The factors of −1-1−1 are ±1\pm 1±1.

STEP 5

List all factors of the leading coefficient −25-25−25.The factors of −25-25−25 are ±1,±5,±25\pm 1, \pm 5, \pm 25±1,±5,±25.

STEP 6

STEP 7

Ensure no duplicate values in the list.The list of possible rational zeros is:±1,±15,±125\boxed{\pm 1, \pm \frac{1}{5}, \pm \frac{1}{25}}±1,±51​,±251​​

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QuestionUse the rational zeros theorem to list all possible rational zeros of the following. $f(x)=-25 x^{3}-2 x^{2}+4 x-1$
Be sure that no value in your list appears more than once.

1. Identify the constant term and the leading coefficient.
2. List all factors of the constant term.
3. List all factors of the leading coefficient.
4. Use the Rational Zeros Theorem to list all possible rational zeros.
5. Ensure no duplicate values in the list.

Identify the constant term and the leading coefficient of the polynomial $f(x) = -25x^3 - 2x^2 + 4x - 1$ .
The constant term $a_0$ is $-1$ .
The leading coefficient $a_n$ is $-25$ .

List all factors of the constant term $-1$ .
The factors of $-1$ are $\pm 1$ .

List all factors of the leading coefficient $-25$ .
The factors of $-25$ are $\pm 1, \pm 5, \pm 25$ .

Ensure no duplicate values in the list.
The list of possible rational zeros is:
$\boxed{\pm 1, \pm \frac{1}{5}, \pm \frac{1}{25}}$