91Ó°ÊÓ

The zeros of a polynomial, sometimes called roots, are the values for which the polynomial equals zero. For instance, if you have a polynomial equation like

• \( p(x) = (x - 2)(x + 2)(x - 1)(x + 1) \)

it means that setting each factor equal to zero, gives you the zeros of the polynomial. These zeros are

• \( x = 2 \) and \( x = -2 \)
• \( x = 1 \) and \( x = -1 \)

These are also called the roots or solutions of the polynomial equation because they are the x-values where the graph intersects the x-axis. Knowing the zeros helps us understand the behavior of the polynomial, especially in relation to its graph and symmetry properties. For example, if these zeros are real and distinct, like in our case, the graph will cross the x-axis at four distinct points.

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. It plays a critical role in determining the end behavior of the polynomial graph. For the polynomial

• \( p(x) = 117(x^4 - 5x^2 + 4) \)

we see that the term with the highest degree is

• \( 117x^4 \)

Therefore, the leading coefficient is 117. This leading coefficient tells us how the polynomial stretches or compresses in the vertical direction. For example, a larger leading coefficient indicates that the graph will be steeper as it moves away from the x-axis toward the ends. In our polynomial's case, multiplying by 117 has increased this steepness dramatically. This property affects the width and orientation of the polynomial's graph. Usually, when the leading coefficient is positive, the graph's ends will point upwards.

The factored form of a polynomial is expressed as a product of its factors. It is a helpful way to understand and identify the zeros of the polynomial, especially when they are easily readable from the equation. For example, if we start with

• \( p(x) = (x - 2)(x + 2)(x - 1)(x + 1) \)

this showcases its factored form directly. This form shows each term being subtracted from x, indicating that these corresponding values are zeros of the polynomial. Using factored form like this is efficient and visually straightforward since it clearly lays out where the polynomial will equal zero—highlighted by those individual factor terms. Furthermore, factored forms allow for easier computations and understanding of multiplication concepts like the distributive property when further operations or simplifications are needed.

The difference of squares is a special algebraic identity and a helpful tool in simplifying polynomials or finding factors. It is based on the principle that any two squares subtracted from one another can be expressed as a product:

• \( a^2 - b^2 = (a - b)(a + b) \)

In our polynomial exercise, we use the difference of squares twice:

• \( x^2 - 4 = (x - 2)(x + 2) \)
• \( x^2 - 1 = (x - 1)(x + 1) \)

Recognizing this pattern allows for quicker factorization, turning what initially seems like a complex polynomial into manageable parts. Knowing and applying this identity is a powerful strategy, simplifying the task of reducing expressions and solving polynomial equations. It highlights a symmetry in polynomial expressions, making problems involving squares less intimidating and more algorithmically solvable.