91Ó°ÊÓ

When working with polynomials, the greatest common factor (GCF) is the highest degree monomial that divides each term of the polynomial. For the polynomial function given, \(n(x) = x^{6} + 4x^{5} + 4x^{4}\), you need to determine the GCF of the terms.

The polynomial terms are: \(x^{6}, 4x^{5}, \) and \(4x^{4}\). The GCF is the term with the lowest power of \(x\), which is \(x^4\). Thus, factoring this out gives:

\[ n(x) = x^{4}(x^{2} + 4x + 4) \]

Extracting the GCF simplifies the polynomial and prepares it for further factorization. Always check for a GCF first to make polynomials easier to handle.

Factoring polynomials involves breaking down a polynomial into simpler polynomials called factors. After factoring out the GCF in \(n(x)\), you are left with a quadratic expression: \(x^{2} + 4x + 4\). Next, you focus on factoring this quadratic polynomial.

To check if a quadratic can be factored easily, look for patterns such as perfect square trinomials or use the quadratic formula if necessary. In this case, notice that \(x^{2} + 4x + 4\) is a perfect square trinomial:

\[ x^{2} + 4x + 4 = (x + 2)^{2} \]

This further reduces the polynomial to:

\[ n(x) = x^{4}(x + 2)^{2} \]

This method makes solving for zeros straightforward as it reduces complex expressions to simpler, more manageable ones.

The multiplicity of a zero in a polynomial function refers to how many times that zero appears as a root of the polynomial. From the factored form of our polynomial, \(n(x) = x^{4}(x+2)^{2}\), we can determine the zeros and their multiplicities.

To find the zeros, set each factor equal to zero:

- For \(x^4 = 0\): The only zero is \(x = 0\). The exponent (4) denotes that this zero has a multiplicity of 4.
- For \((x + 2)^2 = 0\): The zero is \(x = -2\). The exponent (2) indicates that this zero has a multiplicity of 2.

The multiplicity affects the graph of the polynomial. A zero of odd multiplicity (e.g., 1, 3, 5) causes the graph to cross the x-axis at that zero. A zero of even multiplicity (e.g., 2, 4, 6) causes the graph to touch the x-axis and turn around at that zero.

A perfect square trinomial is a quadratic expression of the form \(a^{2} + 2ab + b^{2}\), which can be factored into \((a + b)^{2}\). These trinomials appear frequently in polynomial factorization and make finding roots much easier.

For the quadratic \(x^{2} + 4x + 4\), we can identify it as a perfect square trinomial because:

- The first term is \(x^{2}\), which is \(x\) squared.
- The last term is 4, which is \(2\) squared.
- The middle term is \(4x\), which is \(2 \cdot x \cdot 2\).

Thus, \(x^{2} + 4x + 4 = (x+2)^{2}\).
A visual recognition of such patterns can save time and simplify the factors, making it easier to solve for the polynomial's zeros. This recognition is particularly useful in combination with other polynomial simplification techniques.