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The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, when you plug a zero into the polynomial, the result is zero. For a polynomial expressed as a product of factors, each factor can be set to zero to find its corresponding zero(s). This helps in understanding and visualizing where the polynomial intersects the x-axis on a graph.

In the given exercise, the function is the product of two factors:

• \((2x+1)^3\)
• \(9x^2 - 6x + 1\)

Each factor individually provides zeros.For the first factor, \((2x+1)\), setting it to zero gives the zero at \(x = -\frac{1}{2}\).
The second, quadratic factor \(9x^2 - 6x + 1\), requires the quadratic formula to solve for its zeros, where the discriminant plays an important role.

The multiplicity of a root refers to the number of times that root appears in the polynomial as a zero. This concept helps us determine the behavior of the polynomial at the zero. A root with multiplicity greater than one indicates that the graph touches or is tangent to the x-axis at that root, rather than crossing it.

In the given function, each zero has a specified multiplicity:

• The zero \(x = -\frac{1}{2}\) from the factor \((2x+1)^3\) has a multiplicity of 3, as indicated by the exponent.
• The zero \(x = \frac{1}{3}\) from the quadratic factor has a multiplicity of 2, derived from the quadratic having one unique root root (a double root due to a zero discriminant).

This means that the polynomial will touch the x-axis at \(x=-\frac{1}{2}\) and be tangent (not passing through) at \(x = \frac{1}{3}\). A root with an odd multiplicity generally crosses the x-axis.

The quadratic formula is a fundamental tool for solving quadratic equations, represented as \(ax^2 + bx + c = 0\). It provides the solutions (or zeros) of the equation: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is incredibly useful when factoring is difficult or impossible.

For the second factor of our polynomial \(9x^2 - 6x + 1\), we apply the quadratic formula with

• \(a = 9\)
• \(b = -6\)
• \(c = 1\)

Substituting these values into the formula will yield the zeros of this factor. Understanding how to use this formula offers a reliable method to find zeros, especially in more complex situations where simplification is not straightforward.

The discriminant is a part of the quadratic formula under the square root, given by \(b^2 - 4ac\). It gives essential information about the nature of the roots of a quadratic equation:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If it's zero, there is one real double root (meaning the graph just touches the x-axis once).
• If it's negative, the roots are complex and not real, which means the graph doesn't touch the x-axis.

In the problem, the discriminant for the factor \(9x^2 - 6x + 1\) was zero, indicating a single real root that is repeated, or a double root. This affected the multiplicity of the root \(x = \frac{1}{3}\), emphasizing the importance of calculating the discriminant to understand the solution fully.