91Ó°ÊÓ

The **difference of squares** is a special algebraic pattern that simplifies certain expressions quickly. It is recognized in expressions of the form \(a^2 - b^2\). This pattern emerges because it can always be rewritten as the product of two binomials: \((a - b)(a + b)\).

In our example, the polynomial \(Q(x) = x^4 - 1\) fits this pattern because it can be seen as \((x^2)^2 - (1)^2\). Here, \(a = x^2\) and \(b = 1\). By applying the difference of squares, the polynomial becomes \((x^2 - 1)(x^2 + 1)\).

This method not only simplifies the expression but also helps in quickly identifying the roots by breaking down complex expressions into simpler parts. Note that while \(x^2 - 1\) can be further factored using the same pattern, \(x^2 + 1\) does not factor over the real numbers, indicating the presence of complex solutions.

The **zeros of a polynomial** are the values of \(x\) that make the polynomial equal to zero. These zeros can also be called roots or solutions.

To find these zeros, we set the expression equal to zero and solve. For \(Q(x) = (x-1)(x+1)(x^2 + 1)\), the zeros are found by setting each factor to zero:

• \(x-1 = 0\) leads to \(x = 1\)
• \(x+1 = 0\) leads to \(x = -1\)
• \(x^2 + 1 = 0\) which simplifies to \(x^2 = -1\), resulting in \(x = i\) and \(x = -i\)

The roots \(x = 1\) and \(x = -1\) are real numbers, while \(x = i\) and \(x = -i\) are complex numbers, demonstrating that polynomials of higher degrees can have a mix of real and complex roots.

The **multiplicity of zeros** refers to how many times a particular solution occurs for a polynomial equation. It is related to the degree of the factor in the polynomial.

In the complete factorization \(Q(x) = (x-1)(x+1)(x^2 + 1)\), each factor \((x-1)\), \((x+1)\), and \((x^2 + 1)\) is only raised to the power of one. Therefore, each zero, \(x = 1\), \(x = -1\), \(x = i\), and \(x = -i\), has a multiplicity of 1.

This means each root is simple and only appears once in the solution set. Recognizing multiplicities is essential since it affects the graph's behavior near those roots - for an even multiplicity, the graph will touch and bounce off the x-axis, while for an odd multiplicity, it will cut through the x-axis.