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The multiplicity of a zero in a polynomial refers to how many times a particular zero appears. If \(x = a\) is a zero of a polynomial \(P(x)\), it means that when you substitute \(a\) into the polynomial, the result is zero, implying \(P(a) = 0\). This concept helps us understand the behavior of the polynomial at that zero.
For example, in the case of the polynomial \(P(x) = x^5 + 9x^4 + 33x^3 + 55x^2 + 42x + 12\), we verified that \(-1\) is a zero of multiplicity 3. This means that \(-1\) is a root three times over.

The multiplicity affects the shape of the graph at that point. If the multiplicity is odd, the graph crosses the x-axis at the zero. If it’s even, the graph merely touches the x-axis but doesn’t actually cross it.

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form \(x - c\). It simplifies the division process compared to long division, especially when you repeatedly check for zeros or factor a polynomial.
To perform synthetic division, follow these steps:

• Identify the root \(c\) of the divisor \(x - c\) (in our example, \(c = -1\)).
• Write down the coefficients of the polynomial \(1, 9, 33, 55, 42, 12\).
• Using the root, transform these coefficients step by step.

In the solution, synthetic division was applied three times, each further simplifying the polynomial one degree lower. After three successful divisions, \(-1\) was confirmed as a zero with multiplicity 3. Synthetic division is not only a tool for solving roots but also for understanding the structure and behavior of polynomials.

The quadratic formula is a powerful tool used to find the zeros of quadratic polynomials. A quadratic equation in standard form is given by \(ax^2 + bx + c = 0\). The quadratic formula provides a way to calculate the zeros directly using the coefficients of the quadratic:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The part under the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, there are two complex roots.

In our exercise, the polynomial \(x^2 + 6x + 12\) resulted in a negative discriminant, indicating complex zeros. By applying the quadratic formula, we determined that the complex zeros are \(-3 + i\sqrt{3}\) and \(-3 - i\sqrt{3}\).

Complex zeros arise when the solutions to a polynomial equation include imaginary numbers. This occurs when the discriminant in the quadratic formula is negative, producing an expression under the square root that requires a solution in terms of complex numbers.
Complex numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). In the polynomial from our exercise, after applying the quadratic formula, we found the zeros \(-3 + i\sqrt{3}\) and \(-3 - i\sqrt{3}\).
These roots come in conjugate pairs, which means if \(a + bi\) is a zero of a polynomial with real coefficients, then \(a - bi\) will also be a zero. This symmetry is crucial when factoring polynomials into linear factors, particularly for those of higher degrees.