Questions: Example 15. What are the rational zeros of the following polynomial? 2 x^3 + 7 x^2 + 2 x - 3 possible rational zeros: actual rational zeros:

Transcript text: Example 15. What are the rational zeros of the following polynomial? \[ 2 x^{3}+7 x^{2}+2 x-3 \] possible rational zeros: $\qquad$ actual rational zeros: $\qquad$

Solution

Solution Steps

To find the rational zeros of a polynomial, we can use the Rational Root Theorem. This theorem states that any rational solution, expressed as a fraction $ \frac{p}{q} $, has $ p $ as a factor of the constant term and $ q $ as a factor of the leading coefficient. For the polynomial $ 2x^3 + 7x^2 + 2x - 3 $, the possible rational zeros are the factors of $-3$ divided by the factors of $2$. We then test these possible zeros by substituting them into the polynomial to see which ones yield a result of zero.

Step 1: Identify the Polynomial

We are given the polynomial $ 2x^3 + 7x^2 + 2x - 3 $.

Step 2: Determine Possible Rational Zeros

Using the Rational Root Theorem, we find the possible rational zeros by taking the factors of the constant term $-3$ and the leading coefficient $2$. The possible rational zeros are: \[ \text{Possible Zeros} = \left\{-3, -\frac{3}{2}, -1, -\frac{1}{2}, 1, \frac{1}{2}, 3, \frac{3}{2}\right\} \]

Step 3: Test Possible Zeros

We substitute each possible zero into the polynomial to determine which ones yield a result of zero. The actual rational zeros found are: \[ \text{Actual Zeros} = \{-3, -1, \frac{1}{2}\} \]