To find the real zeros of the polynomial function $f(x) = x^4 + 4x^3 - 9x^2 - 16x + 20$, we can apply several techniques such as factoring, using the Rational Root Theorem, or numerical methods if necessary. Let's start by trying the Rational Root Theorem to find potential rational zeros.

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests that any possible rational root $\frac{p}{q}$, where $p$ divides the constant term (20) and $q$ divides the leading coefficient (1), must be a factor of 20.

Factors of 20: $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$

Since the leading coefficient is 1, the possible rational roots are simply: $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$

Step 2: Test possible roots

Let's try these values in $f(x)$:

• $f(1) = 1^4 + 4(1)^3 - 9(1)^2 - 16(1) + 20 = 1 + 4 - 9 - 16 + 20 = 0$

So, $x = 1$ is a root of the polynomial.

Step 3: Perform synthetic division

Now, we will use synthetic division to divide $f(x)$ by $(x - 1)$.

Performing synthetic division of $x^4 + 4x^3 - 9x^2 - 16x + 20$ by $x - 1$:

1 & 1 & 4 & -9 & -16 & 20 \\ & & 1 & 5 & -4 & -20 \\ \hline & 1 & 5 & -4 & -20 & 0 \\ \end{array}$$ The quotient is $$x^3 + 5x^2 - 4x - 20$$. ### Step 4: Find the zeros of the quotient We now have $$x^3 + 5x^2 - 4x - 20$$. Let's try the Rational Root Theorem again for this cubic polynomial. Possible rational roots are still $$\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$$. - $$f(2) = 2^3 + 5(2)^2 - 4(2) - 20 = 8 + 20 - 8 - 20 = 0$$ So, $$x = 2$$ is a root. ### Step 5: Synthetic division again Divide $$x^3 + 5x^2 - 4x - 20$$ by $$x - 2$$: $$\begin{array}{r|rrrr} 2 & 1 & 5 & -4 & -20 \\ & & 2 & 14 & 20 \\ \hline & 1 & 7 & 10 & 0 \\ \end{array}$$ The quotient is $$x^2 + 7x + 10$$. ### Step 6: Solve the quadratic equation Now solve $$x^2 + 7x + 10 = 0$$ using the quadratic formula: $$x = \frac{-7 \pm \sqrt{7^2 - 4(1)(10)}}{2(1)} = \frac{-7 \pm \sqrt{49 - 40}}{2} = \frac{-7 \pm \sqrt{9}}{2} = \frac{-7 \pm 3}{2}$$ Thus, the solutions are: $$x = \frac{-7 + 3}{2} = -2 \quad \text{and} \quad x = \frac{-7 - 3}{2} = -5$$ ### Step 7: List all real zeros The real zeros of $$f(x) = x^4 + 4x^3 - 9x^2 - 16x + 20$$ are: $$x = 1, \, 2, \, -2, \, -5$$ Would you like more details or clarifications? --- Here are 5 questions related to this topic: 1. What is the Rational Root Theorem and how is it applied to find zeros? 2. How does synthetic division work in polynomial division? 3. How can you solve higher-degree polynomials with multiple real and complex roots? 4. What is the importance of factoring polynomials in solving equations? 5. Can every polynomial be factored into real linear factors? **Tip:** Always check for rational roots first to simplify solving higher-degree polynomials!