Problem 31 - Precalculus with Calculus Previews

Question

Use synthetic division to find the quotient \(q(x)\) and remainder \(r\) when \(f(x)\) is divided by the given linear polynomial. $$ f(x)=x^{3}-(2+\sqrt{3}) x^{2}+3 \sqrt{3} x-3 ; \quad x-\sqrt{3} $$

Step-by-Step Solution

Answer

The quotient is \(x^2 - 2x + \sqrt{3}\) and the remainder is 0.

Step 1: Set up synthetic division

Identify the divisor in the form \(x - c\), where \(c = \sqrt{3}\). Arrange the coefficients of \(f(x)\): \(1, -(2+\sqrt{3}), 3\sqrt{3}, -3\). Prepare to perform synthetic division with \(c = \sqrt{3}\).

Step 2: Start the division process

Bring down the leading coefficient, 1, as it is. Multiply it by \(\sqrt{3}\) and add it to the next coefficient, \(-(2+\sqrt{3})\). This gives: \(1 \times \sqrt{3} = \sqrt{3}\), then add to \(-(2 + \sqrt{3})\) to get: \(-2\).

Step 3: Continue division

Multiply the result from Step 2, \(-2\), by \(\sqrt{3}\), giving \(-2\sqrt{3}\). Add this to the next coefficient, \(3\sqrt{3}\), leading to: \(3\sqrt{3} - 2\sqrt{3} = \sqrt{3}\).

Step 4: Complete the division

Multiply your latest result, \(\sqrt{3}\), by \(\sqrt{3}\), getting 3. Add this to the last coefficient \(-3\). So, \(3 - 3 = 0\).

Step 5: Write quotient and remainder

The synthetic division gives the quotient \(q(x) = x^2 - 2x + \sqrt{3}\) and a remainder of 0, indicating \(x - \sqrt{3}\) divides \(f(x)\) perfectly.

Key Concepts

Polynomial Division

Polynomial division is a key mathematical operation that allows us to divide one polynomial by another, simplifying complex polynomial expressions. Synthetic division is a streamlined version of this process, specifically used when dividing by a linear polynomial, commonly illustrated in the form of \(x - c\). Rather than using traditional long division, which can be cumbersome, synthetic division provides a quick and efficient alternative.

• Identify the divisor, ensuring that it’s in the form \(x - c\).
• Extract the coefficients of the polynomial you wish to divide.
• Perform synthetic division using these coefficients and the value \(c\).

This method is particularly handy for polynomials that are not too high in degree and helps simplify the division process significantly. It's a great tool for quickly determining the quotient and remainder of polynomial division.

Quotient and Remainder

When dividing polynomials, the outcome produces two main results: the quotient and the remainder. The quotient is the result of the division itself, representing the simplified polynomial expression obtained after division. The remainder is what is left over after the division process.

• The quotient is used to represent the divided form of the original polynomial.
• If the remainder is zero, it indicates that the divisor perfectly divides the polynomial.
• A non-zero remainder suggests that the polynomial does not divide perfectly and there are leftover terms.

In our example using synthetic division, the remainder was zero, which means the polynomial \(x - \sqrt{3}\) divides the original polynomial perfectly. The obtained quotient was \(q(x) = x^2 - 2x + \sqrt{3}\). This captures the essence of using synthetic division efficiently to find precise results in polynomial calculations.

Roots of Polynomials

Roots of polynomials are values for which the polynomial expression equals zero. They are crucial in understanding more about the behavior of polynomial functions.

• If \(x - c\) divides a polynomial \(f(x)\) with a remainder of zero, then \(c\) is a root of that polynomial.
• Finding the roots of a polynomial is akin to solving the equation \(f(x) = 0\).
• Roots can be real or complex numbers, depending on the polynomial's structure.

In the given example, since \(x - \sqrt{3}\) divides \(f(x)\) with a remainder of zero, \(\sqrt{3}\) is indeed a root of the polynomial \(f(x) = x^3 - (2+\sqrt{3})x^2 + 3\sqrt{3}x - 3\). Understanding this, we see the importance of identifying roots in polynomial equations as they reveal points where the graph of the polynomial intersects the x-axis.