91Ó°ÊÓ

When dealing with polynomials, especially when solving cubic equations, synthetic division is a powerful tool. It's a shortcut to divide a polynomial by a binomial of the form \(x - k\), where \(k\) is a known root of the polynomial.

To understand synthetic division better, consider your equation: \(3x^3 - 11x^2 + 16x - 12\) with one known root being \(x = 2\). This tells us that the polynomial can be divided by \(x - 2\).

Here is a simplified step-by-step approach of synthetic division:

• Write down the coefficients of the polynomial: \(3, -11, 16, -12\).
• Bring down the leading coefficient, which is \(3\), to the bottom row.
• Multiply \(3\) by the root \(2\) and place it under the next coefficient \(-11\).
• Add \(-11\) and \(6\) (which is \(3 \times 2\)), resulting in \(-5\).
• Repeat this process for each subsequent coefficient until the last term confirms that the polynomial is divisible, resulting in zero.
  
  The process leaves you with a new polynomial, in this case \(3x^2 - 5x + 6\), after confirming that \(x-2\) is indeed a factor.

Complex roots often appear when solving polynomials and occur especially when the discriminant of a quadratic equation is negative. The discriminant is part of the quadratic formula under the square root sign. If this value is less than zero, it indicates the presence of a square root of a negative number, leading us to complex numbers.

In our example, after simplifying the cubic equation using synthetic division, we obtained a quadratic equation: \(3x^2 - 5x + 6 = 0\).

To determine the type of roots of this quadratic, compute the discriminant:
\[b^2 - 4ac = (-5)^2 - 4 \times 3 \times 6 = 25 - 72 = -47\]

Since the discriminant is negative (\(-47\)), this tells us no real solutions exist. The solutions will instead be complex, taking the form of \(a + bi\), where \(i\) is the imaginary unit, \(\sqrt{-1}\). Complex roots always appear in conjugate pairs, ensuring polynomials with real coefficients maintain real number symmetry.

The quadratic formula is a vital tool in algebra for finding the roots of a quadratic equation. It's particularly useful when simpler methods like factoring are challenging. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula calculates the roots of any quadratic equation and helps determine the nature of those roots:

• If the discriminant \(b^2 - 4ac\) is positive, the equation has two distinct real roots.
• If it's zero, there's exactly one real root (or a repeated root).
• For a negative discriminant, as seen in our scenario, the roots become complex.

In our example with \(3x^2 - 5x + 6 = 0\), substituting \(a = 3\), \(b = -5\), \(c = 6\) confirms complex roots because the discriminant is \(-47\). The quadratic formula will show the exact roots, which include imaginary numbers due to the square root of a negative value.