91Ó°ÊÓ

The concept of the difference of squares is a crucial tool in polynomial factorization. It allows us to break down certain expressions into easily manageable products. A difference of squares takes the form \( a^2 - b^2 \). You can factor it using the formula: \( (a-b)(a+b) \). This works because multiplying \( (a-b) \) and \( (a+b) \) reconstructs the original expression by cancelling out the middle terms.
In our exercise, the polynomial \( 16x^4 - 81 \) can be rewritten as \( (4x^2)^2 - 9^2 \). Recognizing this as a difference of squares was the key first step to solving the problem.
Using the formula, we first factor this expression into \( (4x^2 - 9)(4x^2 + 9) \). The next step involves recognizing \( 4x^2 - 9 \) itself as another difference of squares, which can be further broken down to \( (2x-3)(2x+3) \).
The approach allows us to simplify and solve by dealing with smaller, more manageable algebraic expressions. This sequential factorization demonstrates how the technique can be applied repeatedly whenever a part of the polynomial fits the difference of squares form.

Determining the zeros of a polynomial is an essential process in algebra. The zeros are the values of \( x \) where the polynomial equals zero. This essentially answers the question, "What values make the polynomial expression be zero?"
In our example, by fully factoring \( P(x) = 16x^4 - 81 \) into \( (2x-3)(2x+3)(4x^2+9) \), we have set ourselves up to find the zeros easily. We specifically look at each factor and solve for \( x \) when each factor itself equals zero.
This gives us the solutions:

• \( 2x - 3 = 0 \) implies \( x = \frac{3}{2} \).
• \( 2x + 3 = 0 \) implies \( x = \frac{-3}{2} \).
• For \( 4x^2 + 9 = 0 \), solving gives the complex zeros \( x = \frac{3i}{2} \) and \( x = \frac{-3i}{2} \).

This systematic method helps identify all zeros, providing a complete picture of where the polynomial intersects the x-axis on a graph.

The multiplicity of a zero is essentially the count of how many times a particular zero appears in a factorized polynomial. This concept gives insight into the behavior of the polynomial at and around its zeros.
If a zero has a multiplicity greater than one, it suggests that the polynomial "touches" the x-axis at this point, but does not cross it. A zero with multiplicity one indicates a simple crossing point.
In our given polynomial \( P(x) = 16x^4 - 81 \), each zero turned out to have a multiplicity of one, indicating simple crossings. This was determined during the factorization process:

• \( x = \frac{3}{2} \) and \( x = \frac{-3}{2} \) correspond to single occurrences in the linear terms \( (2x-3) \) and \( (2x+3) \).
• The complex zeros \( x = \frac{3i}{2} \) and \( x = \frac{-3i}{2} \) appear once, each related to \( 4x^2 + 9 \).

Understanding multiplicities is vital as it assists in sketching accurate polynomial graphs and interpreting polynomial functions in various contexts.