91Ó°ÊÓ

Polynomial division is a method used to divide one polynomial by another, similar to how we divide numbers. In the case of the polynomial \( f(x) = 2x^3 + 3x^2 + x + 6 \) divided by the factor \( x + 2 \), we use either synthetic division or long division. These methods help simplify the original polynomial into a form where we can easily identify factors and zeros.

• In our problem, synthetic division was chosen because it is faster and more straightforward for polynomials with linear divisors.
• We started the process by converting the divisor \( x + 2 \) into the value \( x = -2 \) because we solve at \( x = -a \) where \( a \) is the number attached to \( x \) in the divisor.

Using synthetic division:

• We lined up the coefficients of the polynomial: \( 2, 3, 1, 6 \).
• Brought down the first coefficient, \( 2 \), then multiplied it by \( -2 \), resulting in \( -4 \).
• Added \( -4 \) to the next coefficient \( 3 \) to get \( -1 \), and repeated this process until reaching the end.
• The result was the quadratic \( 2x^2 - x + 3 \), which had to be factored further or solved using another method.

When a quadratic expression does not easily factor, like \( 2x^2 - x + 3 \) from our polynomial division, the quadratic formula becomes our powerful tool. It helps find the roots or zeros of a quadratic equation \( ax^2 + bx + c = 0 \). The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation from our example, \( a = 2 \), \( b = -1 \), and \( c = 3 \). Plugging these values into the formula, we start by calculating the discriminant:

\( b^2 - 4ac = (-1)^2 - 4 \times 2 \times 3 = 1 - 24 = -23 \)

A negative discriminant indicates that the quadratic equation has no real solutions, only complex ones. Therefore, the quadratic part \( 2x^2 - x + 3 \) does not contribute additional real zeros to our original polynomial. In real-world applications, this negative discriminant tells us the curve does not cross the x-axis at any other point except those found from other factors.

Real zeros of a polynomial are the \( x \)-values for which the polynomial equals zero. They correspond to the points where the graph intersects the x-axis. Finding real zeros is fundamental in polynomial graphing and helps us understand the behavior of functions in calculus and algebra.

• The Factor Theorem states that for \( f(x) = 2x^3 + 3x^2 + x + 6 \), \( x+2 \) is a factor if \( f(-2) = 0 \).
• Substituting \( x = -2 \) provides \( f(-2) = 0 \), which confirms that \( x = -2 \) is a real zero of the polynomial.

In this exercise:
• The only real zero is found from this factorization method, which ends at \( x=-2 \) due to no other real solutions found by the quadratic part \( 2x^2 - x + 3 \).
Real zeros play a crucial role in analyzing the roots of polynomials as they describe the x-intercepts of the graph. Remember, every polynomial equation equates to the expression: factors multiplied together equals zero, guiding us toward both real and complex zeros.