In Exercises 63–64, find all the zeros of each polynomial function and write the polynomial as a

step 1 We have the function G of x equals x to the fourth minus 4x cubed plus 8x squared minus 16x plus

In Exercises 63–64, find all the zeros of each polynomial...

Key Concepts

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero occurs for a polynomial. A zero with a multiplicity greater than one indicates that the corresponding linear factor is repeated in the factorization of the polynomial. Understanding the multiplicity of zeros is important because it affects the graph of the polynomial; for example, zeros with odd multiplicities generally indicate that the graph crosses the x-axis, whereas even multiplicities suggest that the graph touches the x-axis and turns around.

Polynomial Zeros

Polynomial zeros, also known as roots, are the values of x for which the polynomial evaluates to zero. They represent the x-intercepts of the polynomial function when graphed on the Cartesian plane. Understanding and finding these zeros is essential because they are used to break down the polynomial into simpler components, and they reveal important properties of the function such as its behavior and the number of times it touches or crosses the x-axis.

Factorization of Polynomials

Factorization of polynomials involves expressing a polynomial as a product of its factors, which can include linear factors corresponding to its zeros. This process simplifies the polynomial and makes it easier to analyze and solve. Factoring is a central concept in algebra and is used to reveal the structure of the polynomial, investigate divisibility, and solve polynomial equations by setting each factor equal to zero.

Rational Root Theorem

The Rational Root Theorem provides a method to identify the possible rational zeros of a polynomial with integer coefficients. It states that any rational zero, expressed in simplest form p/q, divides the constant term and the leading coefficient respectively. This theorem is a crucial tool in polynomial factorization, as it narrows down the candidates for actual zeros that can be verified through substitution or division.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form (x - c). It is particularly useful for testing potential zeros quickly, as it simplifies the division process compared to long division. Synthetic division not only helps confirm whether a suspected zero is valid but also aids in reducing the degree of the polynomial for further factorization.

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