$$ \leftarrow $$ Use the Rational Zero Theorem to list all possible rational zero: $$ f(x)=-3 x^{4}+16 x^{3}+17 x^{2}-17 x+6 $$

Question

UpStudy AI · Accepted Answer

To use the Rational Zero Theorem, we start by noting that any possible rational zero for the polynomial

f(x) = -3x⁴ + 16x³ + 17x² - 17x + 6

must be of the form p/q, where:

• p is a factor of the constant term, 6,
 • q is a factor of the leading coefficient, -3.

Step 1. List the factors:

• Factors of the constant term 6 are: ±1, ±2, ±3, ±6.
 • Factors of the leading coefficient -3 are: ±1, ±3.

Step 2. Write down all possible fractions p/q:

Using q = 1:
  p/1 gives: ±1, ±2, ±3, ±6.

Using q = 3:
  p/3 gives: ±1/3, ±2/3, ±3/3, ±6/3.
  Note that ±3/3 simplifies to ±1 and ±6/3 simplifies to ±2, which are already listed.

Step 3. Combine the unique values:

The possible rational zeros are:
  ±1, ±2, ±3, ±6, ±1/3, ±2/3.

Thus, by the Rational Zero Theorem, the list of all possible rational zeros for f(x) is:

1, -1, 2, -2, 3, -3, 6, -6, 1/3, -1/3, 2/3, -2/3.
The possible rational zeros are: ±1, ±2, ±3, ±6, ±1/3, ±2/3.

\( \leftarrow \) Use the Rational Zero Theorem to list all possible rational zero: \[ f(x)=-3 x^{4}+16 x^{3}+17 x^{2}-17 x+6 \]