91Ó°ÊÓ

A polynomial expressed in factored form makes it much easier to identify the zeros, or roots, of the function. Consider the polynomial given: To express it in factored form, the key step is recognizing parts of the polynomial that can be rewritten in a more manageable way. In our example, we used a substitution method where we let a new variable, such as \( y = x^3 \), simplify our polynomial to a quadratic form.

• This transformation helps to recognize the polynomial as a perfect square trinomial.

Once in factored form, \[ (y - 1)^2 = 0 \] it becomes apparent how this expression yields the zero of the polynomial, in this case, \( y = 1 \). Learning to rewrite polynomials in their factored form is a crucial algebraic skill that eases finding zeros and sketching graphs of polynomial functions.

Zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. In our exercise, the polynomial is set to zero, giving:\[ x^6 - 2x^3 + 1 = 0 \]The substitution \( y = x^3 \) changes the polynomial to a quadratic form, \( y^2 - 2y + 1 = 0 \). Factoring it to \( (y-1)^2 = 0 \) reveals the zero of the quadratic equation:

\( y = 1 \)

Since \( y = x^3 \), substituting back gives \( x^3 = 1 \). Solving for \( x \) involves taking the cube root to find \( x = 1 \).

• Thus, \( x = 1 \) is a zero of the original polynomial.

Identifying zeros is integral to graphing polynomials. They indicate where the function intersects or touches the x-axis.

The multiplicity of a root refers to the number of times that root appears as a factor in the polynomial. When a root repeats, as seen in this problem,

• its higher multiplicity implies distinctive behavior at that zero.

For the polynomial \( (y-1)^2 = 0 \), the factor \( (y-1) \) repeats, indicating a multiplicity of 2 for \( y = 1 \). Substituting back gives \( x = 1 \), with a multiplicity of 2. This higher multiplicity affects how the graph behaves at that point:

• At \( x = 1 \), the graph only touches the x-axis and then bounces back.

Higher multiplicities change the flatness at the roots. Understanding the multiplicity aids in accurately sketching polynomial graphs and predicting its graphical behavior.