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Factorization is the process of breaking down an expression into simpler parts, or 'factors,' that when multiplied together give the original expression. In the given polynomial function \_f(x) = (x-2)^3 (x^2-7)_, we can clearly see two factors: \((x-2)^3\) and \(x^2-7\).

Each factor represents a significant part of the polynomial and helps in finding the zeros. Factorization often involves isolating the simplest form of the equation, such as converting \(x^2-7\) into \((x- \sqrt{7})(x+ \sqrt{7})\).

This makes solving the equation easier and more straightforward.

Understanding factorization is crucial since it simplifies polynomials into manageable pieces and reveals the individual roots or zeros of the polynomial function.

Multiplicity refers to the number of times a specific zero appears in a polynomial. This is determined by the exponent of the corresponding factor in the factored form of the polynomial.

For our polynomial function \(f(x)=(x-2)^3 (x^2-7)\), let's inspect each zero:

• The factor \((x-2)^3\) results in the zero \(x=2\) appearing three times, hence having a multiplicity of 3.
• The factor \((x^2-7)\), or more precisely its split form \((x- \sqrt{7})(x+ \sqrt{7})\), results in the zeros \(x= \sqrt{7}\) and \(x= -\sqrt{7}\), each appearing only once, thus each has a multiplicity of 1.

Identifying the multiplicity helps in understanding the behavior of the polynomial at those points. For instance, a zero with a higher multiplicity indicates that the polynomial will touch or bounce off the x-axis at that point, depending on whether the multiplicity is even or odd.

Solving quadratic equations is a fundamental skill for finding the zeros of polynomial expressions, particularly for second-degree polynomials of the form \(ax^2 + bx + c = 0\).

In the given polynomial function, we encounter \((x^2-7)\) which is a simple quadratic equation. To solve it, we set it equal to zero: \[ x^2 - 7 = 0. \]

Next, isolate \x^2\ by adding 7 to both sides: \[ x^2 = 7 \]

To find the values of \x\, we take the square root of each side: \[ x = \pm \sqrt{7} \]

The quadratic equation yields two solutions, \(x=\sqrt{7}\) and \(x=-\sqrt{7}\).

Familiar methods for solving quadratic equations include factoring, completing the square, and using the quadratic formula \(-b \pm \sqrt{b^2-4ac} / 2a\). Mastering these techniques is essential for higher-level algebra and beyond.