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Synthetic division is a simplified method for dividing polynomials, particularly useful when you need to evaluate potential rational zeros, as seen in polynomial functions. It is a shortcut to the more cumbersome polynomial long division and is ideal for dividing a polynomial by a linear expression of the form \( x - c \). Here's how synthetic division streamlines the process:

To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial in order.
• Place the potential zero (from the list of rational zeros) to the left.
• Drop down the leading coefficient.
• Multiply the dropped number by the potential zero and add this to the next coefficient.
• Continue this multiply-and-add process for all coefficients.
• The final number in this operation is the remainder. If it is zero, the tested number is a root of the polynomial.

For example, when testing \( x = 2 \) as a zero of the polynomial \( 2x^3 - 9x^2 + 7x + 6 \), the remainder was zero, confirming it as a zero. Synthetic division thus simplifies both checking and reducing polynomials to lower degrees.

The quadratic formula is an essential tool in algebra, used to find the zeros of quadratic equations, given in the form \( ax^2 + bx + c = 0 \). The solutions for \( x \) can be found using:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula works for all quadratic equations and can also prove useful when factoring seems challenging or impossible.

In the step-by-step solution, once we reduced the polynomial using synthetic division to the quadratic \( 2x^2 - 5x - 3 \), we applied the quadratic formula to find the remaining zeros. With \( a = 2 \), \( b = -5 \), and \( c = -3 \), substituting these into the formula confirmed the zeros as \( x = 3 \) and \( x = -\frac{1}{2} \).

The quadratic formula not only provides solutions directly but also underscores the discriminant \( b^2 - 4ac \), helping determine the nature of the roots (real or complex). In our problem, a positive discriminant indicated two real solutions.

Factoring polynomials involves rewriting them as a product of simpler polynomials. By finding factors, you can solve for zeros efficiently. When the polynomial \( 2x^2 - 5x - 3 \) is factored, it breaks down into simpler linear components, making it easier to find its zeros.

Here's a brief approach to factoring quadratics:

• Try to express the quadratic in the form \( ax^2 + bx + c \) as \( (px + q)(rx + s) \) such that \( p \cdot r = a \) and \( q \cdot s = c \).
• If possible, find two numbers that multiply to \( ac \) and add to \( b \).
• Use these numbers to split the middle term and factor by grouping.

In our example, direct factoring or using the quadratic formula provides the zeros without needing trial and error with factor pairs. Factoring allows you to simplify expressions and solve for roots quickly. It's a key concept for understanding polynomial behavior and root calculation.