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Polynomial long division is a method used to divide a polynomial by another polynomial of lesser or equal degree. It's similar to the long division performed with numbers. The main objective is to determine the quotient and remainder when performing this division. Here, our goal is to divide

• The dividend: the original polynomial, which in this case is \( x^4 + 4x^3 - 2x^2 - 12x + 9 \).
• The divisor: derived from the root's multiplicity, given as \((x - 1)^2 = x^2 - 2x + 1\).

In the process, we:

• Compare the highest degree terms of the dividend and the divisor.
• Divide the leading term of the dividend by the leading term of the divisor to form part of the quotient.
• Multiply the entire divisor by this quotient term and subtract from the original polynomial.
• Repeat the process with the new polynomial (without the subtracted part) until reaching a remainder of zero or a degree lower than the divisor.

Successfully performing polynomial long division here reveals the quotient \( x^2 + 6x + 9 \). This quotient is a crucial step towards fully factoring the polynomial.

Multiplicity of a root refers to the number of times a root, or solution, appears in a polynomial equation. If a polynomial has a root with a multiplicity greater than one, it means that the solution repeats that number of times. In our exercise, the given root \( x = 1 \) is noted to have a multiplicity of 2.

This implies several things:

• The factor associated with \( x = 1 \) appears twice in the polynomial when expressed as product of its factors, hence \((x - 1)^2\).
• Graphically, the curve of the polynomial touches the x-axis at this zero and turns back, indicating that it bounces off the axis at \( x = 1 \).

Understanding multiplicity is valuable because it helps us better describe the behavior of the polynomial function at its roots, as well as aids in the factorization process. It was foundational in determining one of the factors in the polynomial \( P(x) = (x - 1)^2 (x + 3)^2 \).

A perfect square trinomial is a specific type of polynomial that results from squaring a binomial. Recognizing and factoring these trinomials can simplify finding the roots of a polynomial. From the quotient of our long division, we have the trinomial expression \( x^2 + 6x + 9 \).

This expression fits the form of a perfect square trinomial because:

• The first term \(x^2\) is \((x)^2\).
• The last term \(9\) is \((3)^2\).
• The middle term \(6x\) is two times the product of 3 and x, that is, \(2 \times 3 \times x\).

This trinomial can be rewritten as \((x + 3)^2\). Recognizing perfect square trinomials allows us to factor polynomials quickly and easily, simplifying our overall polynomial into a product of linear factors. In this context, it translates the remaining part of the polynomial into \((x + 3)^2\), leading to the full factorized form \((x - 1)^2 (x + 3)^2\).