We can find the rational zeros of the polynomial $P(x) = 2x^3 + 7x^2 + 4x - 4$ using the Rational Root Theorem and factoring the polynomial.

Step 1: Rational Root Theorem

The Rational Root Theorem tells us that any rational root, in the form $\frac{p}{q}$, is a factor of the constant term (in this case, $-4$) divided by a factor of the leading coefficient (in this case, $2$).

• Factors of $-4$: $\pm 1, \pm 2, \pm 4$
• Factors of $2$: $\pm 1, \pm 2$

Thus, the possible rational roots are:

$\frac{p}{q} = \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{4}{2} = \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}$

Step 2: Test the possible rational roots

We test these possible rational roots by substituting them into the polynomial $P(x) = 2x^3 + 7x^2 + 4x - 4$ until we find one that results in zero.

• Test $x = -2$:

$P(-2) = 2(-2)^3 + 7(-2)^2 + 4(-2) - 4 = 2(-8) + 7(4) + 4(-2) - 4 = -16 + 28 - 8 - 4 = 0$

So, $x = -2$ is a root.

Step 3: Factor the polynomial

Since $x = -2$ is a root, $(x + 2)$ is a factor of $P(x)$. We can now perform synthetic or polynomial division to divide $P(x)$ by $(x + 2)$.

Perform synthetic division:

-2 & 2 & 7 & 4 & -4 \\ & & -4 & -6 & 4 \\ \hline & 2 & 3 & -2 & 0 \\ \end{array}$$ The quotient is $$2x^2 + 3x - 2$$, so we have: $$P(x) = (x + 2)(2x^2 + 3x - 2)$$ ### Step 4: Factor the quadratic Now, we factor $$2x^2 + 3x - 2$$. Using the method of factoring quadratics: $$2x^2 + 3x - 2 = (2x - 1)(x + 2)$$ Thus, the fully factored form of the polynomial is: $$P(x) = (x + 2)^2(2x - 1)$$ ### Step 5: Find the rational zeros The rational zeros are the values of $$x$$ that make the factors equal to zero: $$x + 2 = 0 \quad \Rightarrow \quad x = -2$$ $$2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}$$ Thus, the rational zeros of $$P(x)$$ are $$x = -2, -2, \frac{1}{2}$$ (including the repetition of $$-2$$). ### Final Answer: - **Rational zeros**: $$-2, -2, \frac{1}{2}$$ - **Factored form**: $$P(x) = (x + 2)^2(2x - 1)$$ Would you like a detailed breakdown of any part of the process? Here are some related questions: 1. What is the Rational Root Theorem, and how does it help in finding zeros? 2. How can synthetic division be used to factor polynomials? 3. What is the relationship between zeros and factors of a polynomial? 4. How do you factor a quadratic expression like $$2x^2 + 3x - 2$$? 5. Can you use another method to find zeros of a cubic polynomial? **Tip**: Always check for repeated roots by considering the powers of the factors in the factored form of the polynomial!