The Rational Zero Theorem provides a way to identify possible rational zeros of a polynomial. It states that all possible rational zeros are of the form:

$\text{Possible zeros} = \pm \frac{\text{factors of constant term}}{\text{factors of leading coefficient}}.$

Given the polynomial:

$f(x) = x^4 + 10x^3 - 20x^2 - 90x + 99$

Step 1: Identify the constant term and leading coefficient

• Constant term: $99$
• Leading coefficient: $1$

Factors of $99$: $\pm 1, \pm 3, \pm 9, \pm 11, \pm 33, \pm 99$
Factors of $1$: $\pm 1$

Thus, the possible rational zeros are:

$\pm 1, \pm 3, \pm 9, \pm 11, \pm 33, \pm 99$

Step 2: Test possible zeros using synthetic division

We test these values to determine which are zeros of the polynomial.

Test $x = 1$:

Using synthetic division:

1 & 1 & 10 & -20 & -90 & 99 \\ & & 1 & 11 & -9 & -99 \\ \hline & 1 & 11 & -9 & -99 & 0 \\ \end{array}$$ The remainder is $$0$$, so $$x = 1$$ is a zero. ### Step 3: Factorize the polynomial Since $$x = 1$$ is a zero, we can write: $$f(x) = (x - 1)(x^3 + 11x^2 - 9x - 99)$$ Now we factor the cubic polynomial $$x^3 + 11x^2 - 9x - 99$$. #### Test $$x = -3$$: Using synthetic division: $$\begin{array}{r|rrrr} -3 & 1 & 11 & -9 & -99 \\ & & -3 & -24 & 99 \\ \hline & 1 & 8 & -33 & 0 \\ \end{array}$$ The remainder is $$0$$, so $$x = -3$$ is another zero. Thus, we can factor further: $$f(x) = (x - 1)(x + 3)(x^2 + 8x - 33)$$ ### Step 4: Solve the quadratic $$x^2 + 8x - 33 = 0$$ Using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad a = 1, \, b = 8, \, c = -33$$ $$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-33)}}{2(1)}$$ $$x = \frac{-8 \pm \sqrt{64 + 132}}{2}$$ $$x = \frac{-8 \pm \sqrt{196}}{2}$$ $$x = \frac{-8 \pm 14}{2}$$ $$x = 3 \quad \text{or} \quad x = -11$$ ### Final Answer: All real zeros The real zeros of $$f(x)$$ are: $$x = 1, \, x = -3, \, x = 3, \, x = -11$$ Would you like me to provide further steps or verify by multiplying back the factors? Here are five related questions to deepen your understanding: 1. What are the benefits of using the Rational Zero Theorem in polynomial factorization? 2. Can a polynomial have more rational zeros than its degree? Why or why not? 3. How do imaginary zeros arise in polynomials, and how are they paired? 4. What happens if no rational zeros are found for a given polynomial? 5. Can the Rational Zero Theorem help with polynomials with fractional coefficients? **Tip**: Always verify all potential rational zeros and use synthetic division systematically to avoid missing any factors.