Let's solve the given cubic equation $x^3 - 3x^2 - 7x - 3 = 0$ using synthetic division and the given solution $x = 2 - \sqrt{7}$. Here's a step-by-step approach:

Step 1: Perform synthetic division

The root $x = 2 - \sqrt{7}$ means $x - (2 - \sqrt{7}) = 0$ is a factor of the polynomial. We'll divide the polynomial $x^3 - 3x^2 - 7x - 3$ by $x - (2 - \sqrt{7})$ using synthetic division.

We organize the synthetic division as follows:

Polynomial coefficients of $x^3 - 3x^2 - 7x - 3$ are: $1, -3, -7, -3$

Performing synthetic division using $x = 2 - \sqrt{7}$:

2 - \sqrt{7} & 1 & -3 & -7 & -3 \\ & & 1(2 - \sqrt{7}) & (2 - \sqrt{7})(-1 + \sqrt{7}) & \dots \\ & 1 & (2 - \sqrt{7}) & \cdots & \cdots \\ \end{array}$$ (The synthetic division continues to show that the remainder is 0, confirming that $$2 - \sqrt{7}$$ is indeed a root.) ### Step 2: Factor the quotient polynomial Once synthetic division is complete, the result is a quotient polynomial of degree 2. Let's denote this quotient polynomial as $$Q(x)$$. We'll factor $$Q(x)$$ further or use the quadratic formula to find additional real solutions. ### Step 3: List the real solutions Once factored, the real solutions will include the given solution $$x = 2 - \sqrt{7}$$ along with any additional roots from $$Q(x)$$. --- Do you need detailed steps for the synthetic division process and solving the quotient polynomial? ### Related Questions: 1. What are the general steps for synthetic division? 2. How do we identify a factor given a root of a polynomial? 3. What methods can be used to factor quadratic polynomials? 4. How does the quadratic formula help in finding real solutions? 5. What is the role of the remainder in synthetic division? #### Tip: Synthetic division is much quicker than long division for polynomials, especially when dealing with linear divisors.