91Ó°ÊÓ

When we talk about the roots of a polynomial, we're referring to the values of \( x \) that make the polynomial equal to zero. Consider the polynomial \( f(x) \). If \( f(c) = 0 \), then \( c \) is a root of the polynomial \( f(x) \). These roots are crucial because they give us insight into the behavior and characteristics of the polynomial.

• Real Roots: These are the solutions that can be plotted directly on the real number line. For example, if 1 is a root, \( f(1) = 0 \).
• Complex Roots: These can include imaginary numbers, but aren't relevant in this exercise since we're only dealing with real numbers.

Identifying roots helps us both in graphing polynomials and in simplifying them. In this exercise, we know the roots and their multiplicities: 1 occurs twice, and -3 occurs once. Each of these roots contributes a factor to the polynomial.

The term 'multiplicity' refers to the number of times a particular root appears in a polynomial. In simpler terms, it's the exponent of the factor associated with that root in the polynomial expression. For instance, if a root has a multiplicity of 2, it means the factor occurs twice in the factorized version of the polynomial.

• Multiplicity of 1: When a root has multiplicity 1, it only appears once in the polynomial. An example is the root -3 in our problem, which corresponds to the factor \((x + 3)\).
• Multiplicity greater than 1: A root of higher multiplicity, like 1 with multiplicity 2 in our exercise, results in the factor \((x - 1)^2\). The polynomial graph "touches" the x-axis at this root.

The multiplicity affects not only the mathematical expression but also the graph's appearance at each root. High multiplicity can create a "flattening" effect on the curve.

Writing polynomials in standard form means expressing them as a sum of terms, each consisting of a coefficient and a variable raised to a power. The terms are arranged in descending order of power from left to right. For example, the standard form looks like \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \).

• Leading Coefficient: This is the coefficient of the term with the highest power. In our solution, it's 1 for the term \( x^3 \).
• Descending Exponents: The powers of \( x \) are sorted from the highest (\( x^3 \)) to the lowest (constant term \( x^0 \)).

It's important to express the polynomial in standard form, especially when the degree is less than 7, as it neatly organizes the information about the polynomial's behavior and roots. In the exercise, expanding \((x - 1)^2 (x + 3)\) into \( x^3 + x^2 - 5x + 3 \) offers a clear view of its structure and confirms that it complies with the standard form, with a leading coefficient of 1.