91Ó°ÊÓ

Polynomial division is a fundamental algebraic process that helps us break down complex polynomials into simpler parts. This process is similar to long division with numbers, but it involves variables.

When given a polynomial like the one in this exercise, dividing by a factor such as \( x + 2 \) allows us to simplify the original polynomial into smaller, more manageable pieces. In this scenario, we use "synthetic division," a shorthand method that simplifies the process when dividing by linear factors. It's crucial to write down all coefficients of the polynomial in sequence and methodically apply the division.

• First, identify the coefficients of the polynomial.
• Use the corresponding value from the factor as the divisor.
• Process each term step-by-step, ensuring careful arithmetic to avoid mistakes.

The key takeaway is that polynomial division allows us to find other factors or zeros of the polynomial, simplifying the process of finding solutions.

Real zeros of a polynomial are the values of \( x \) where the polynomial equals zero. These zeros may also be referred to as roots or solutions.

Finding the real zeros of a polynomial is often the final goal when solving polynomial equations, as these zeros represent the points where the graph of the polynomial crosses the x-axis.

• Start by using any known factors and substituting corresponding zeros.
• The Factor Theorem is particularly useful as it guarantees that if \( (x + a) \) is a factor, then \( x = -a \) is a real zero.

Once identified, real zeros can simplify problem solving maneuvers, such as constructing graphs or understanding behavior of the polynomial function itself. In our exercise, the real zero was found to be \( x = -2 \).

The quadratic formula is a robust tool for finding the roots of quadratic equations, typically expressed in the form of \( ax^2 + bx + c = 0 \).

In the current exercise, after performing polynomial division, we were left with a quadratic equation \( 2x^2 - x + 3 = 0 \). To find the zeros of this polynomial, we use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Each component of this formula plays a critical role:

• \( -b \) represents the opposite of the quadratic coefficient.
• \( \pm \sqrt{b^2 - 4ac} \) is key for determining the nature of the roots; the term \( b^2 - 4ac \) is called the discriminant.

For this quadratic, the discriminant (\(-23\)) was negative, indicating the absence of real solutions. Hence, the polynomial has complex roots, and no additional real zeros were found, besides the root provided by the Factor Theorem.