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Synthetic division is a simplified method used for dividing a polynomial by a binomial of the form \(x - c\). This technique is often quicker and less error-prone compared to traditional long division. To start synthetic division, you only need the coefficients of the polynomial you're dividing. Let’s say you're working with the polynomial \(x^3 - 6x^2 + 9x - 2\) and dividing by \(x - 2\).

• Write down the coefficients: \(1, -6, 9, -2\).
• Place the root of \(x - 2\) which is \(2\) outside the division symbol.
• Bring the first coefficient, \(1\), down.
• Multiply \(2\) by \(1\) (the value at the bottom row) and write it under the next coefficient.
• Add these two numbers: \(-6 + 2 = -4\) and repeat this process for the next coefficients.

The process continues until you complete the division resulting in a new polynomial quotient and a remainder, if any. Here, the new polynomial is \(x^2 - 4x + 1\), and since the division was exact, the remainder is zero.

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This expression allows us to resolve the roots even if the quadratic cannot be factored easily.
For the quadratic \(x^2 - 4x + 1\), let's use the quadratic formula:

• Identify the coefficients: here, \(a = 1\), \(b = -4\), and \(c = 1\).
• Substitute them into the formula.
• Calculate the discriminant \((b^2 - 4ac)\): this equals \(16 - 4 = 12\).
• Find the roots: \(x_1 = \frac{4 + \sqrt{12}}{2}\) and \(x_2 = \frac{4 - \sqrt{12}}{2}\).

These roots are the solutions to the quadratic equation and are also roots of the original polynomial after division.

Factoring polynomials is an essential skill for solving polynomial equations. By expressing a polynomial as a product of its factors, we identify the values that make the polynomial equal to zero. In this exercise, because \(x - 2\) is already known as a factor, we began by dividing the cubic polynomial \(x^3 - 6x^2 + 9x - 2\) by it.After division, the remaining polynomial \(x^2 - 4x + 1\) was obtained. While this did not readily factor into simpler polynomials, understanding the process of trying to factor can still be valuable:

• Check for common factors among the terms.
• For quadratics like \(ax^2 + bx + c\), factorization involves finding two numbers that multiply to \(ac\) and add to \(b\).
• Often, completing the factorization involves using techniques like synthetic division or the quadratic formula when simple factoring fails.

Factoring is the simplest form of solving polynomials as it directly identifies the zeros, provided the polynomial is not too complex. With \(x^2 - 4x + 1\), using the quadratic formula was necessary to find the roots because straightforward factoring methods weren't applicable.