Math  /  Algebra

QuestionUse the rational zeros theorem to list all possible rational zeros of the following f(x)=7x45x3+9x25x3f(x)=7 x^{4}-5 x^{3}+9 x^{2}-5 x-3
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 f(x)=anxn+an1xn1++a1x+a0 f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 is of the form pq \frac{p}{q} , where p p is a factor of the constant term a0 a_0 and q q is a factor of the leading coefficient an a_n .
2. We need to list all possible rational zeros for the polynomial f(x)=7x45x3+9x25x3 f(x) = 7x^4 - 5x^3 + 9x^2 - 5x - 3 .

STEP 2

1. Identify the constant term and the leading coefficient.
2. Determine the factors of the constant term.
3. Determine the factors of the leading coefficient.
4. List all possible rational zeros using the Rational Zeros Theorem.
5. Ensure the list of possible rational zeros contains no duplicates.

STEP 3

Identify the constant term and the leading coefficient of the polynomial f(x)=7x45x3+9x25x3 f(x) = 7x^4 - 5x^3 + 9x^2 - 5x - 3 .
The constant term a0 a_0 is 3-3.
The leading coefficient an a_n is 77.

STEP 4

Determine the factors of the constant term 3-3.
The factors of 3-3 are ±1,±3\pm 1, \pm 3.

STEP 5

Determine the factors of the leading coefficient 77.
The factors of 77 are ±1,±7\pm 1, \pm 7.

STEP 6

List all possible rational zeros using the Rational Zeros Theorem. These are of the form pq\frac{p}{q}, where pp is a factor of the constant term and qq is a factor of the leading coefficient.
Possible rational zeros are:
±11,±31,±17,±37\pm \frac{1}{1}, \pm \frac{3}{1}, \pm \frac{1}{7}, \pm \frac{3}{7}

STEP 7

Ensure the list of possible rational zeros contains no duplicates.
The unique possible rational zeros are:
±1,±3,±17,±37\pm 1, \pm 3, \pm \frac{1}{7}, \pm \frac{3}{7}
The list of all possible rational zeros is: ±1,±3,±17,±37 \boxed{\pm 1, \pm 3, \pm \frac{1}{7}, \pm \frac{3}{7}} .

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