Problem 1 - College Algebra

Question

In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-x^{3}+x^{2}-4 x-4 $$

Step-by-Step Solution

Answer

The possible rational zeros for the function are \(\pm1, \pm2, and \(\pm4\).

Step 1: Identify the coefficients

Identify the leading coefficient and the constant term in the function. The leading coefficient is the coefficient of \(x^{3}\) which is 1, and the constant term is -4.

Step 2: List the factors

List all the factors of the constant term and the leading coefficient. The factors of -4 are \(\pm1, \pm2, \pm4\) and the factors of 1 are \(\pm1\).

Step 3: Form the ratio

Form all possible ratios using the factors of the constant term and the leading coefficient. As our leading coefficient's factor is only \(\pm1\), our possible rational zeros are the factors of our constant, therefore are \(\pm1, \pm2, \pm4\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression that involve sums of powers of variables, each multiplied by a coefficient. They take the general form: \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\] where:

• \(n\) is a non-negative integer called the degree of the polynomial.
• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients.
• \(x\) is the variable, often representing input values.

In the example \(f(x) = -x^3 + x^2 - 4x - 4\), the polynomial is of degree 3 because the highest power is \(x^3\). Polynomial functions can be easily graphed, and their shape and behavior can be determined by the degree and the coefficients. Understanding the structure of polynomial functions helps in finding roots, analyzing trends, and solving equations, which are key in many mathematical applications.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest power. It's an essential part of understanding a polynomial's behavior, particularly for large values of \(x\). In polynomial expressions, the leading term and its coefficient can have significant implications for the following:

• End behavior: Determines how the polynomial behaves as \(x\) approaches positive or negative infinity.
• Shape of the graph: Influences its symmetry and the direction the graph faces.

For example, in the polynomial function \(f(x) = -x^3 + x^2 - 4x - 4\), the leading coefficient is -1, meaning the polynomial is a cubic function with a negative sign. This tells us that as \(x\) goes very positive or negative, the function will tend to negative infinity. Identifying the leading coefficient is crucial for the Rational Zero Theorem, as it helps form possible rational zeros.

Constant Term

The constant term in a polynomial is the term without a variable, essentially the number part of the polynomial when \(x = 0\). In mathematical terms, if a polynomial is written as \(f(x) = a_nx^n + \ldots + a_1x + a_0\), then \(a_0\) is the constant term.The constant term influences:

• The y-intercept of the graph: This is the point on the graph where it crosses the y-axis (where \(x=0\)).
• Potential rational zeros via the Rational Zero Theorem, as factors of this term are paired with factors of the leading coefficient to find possible zeros.

In the polynomial \(f(x) = -x^3 + x^2 - 4x - 4\), the constant term is -4. To find which rational numbers could be zeros of the polynomial, we consider the factors of -4. This term plays a vital role in applying the Rational Zero Theorem.

Factorization

Factorization in the context of polynomials involves expressing the polynomial as a product of simpler polynomials or factors. This is key for solving polynomial equations and finding zeros (roots). To factor a polynomial means to write it as:\[f(x) = (x - r_1)(x - r_2) \ldots (x - r_n)\]where \(r_1, r_2, \ldots, r_n\) are the roots of the polynomial or values of \(x\) for which \(f(x) = 0\).The ability to factor effectively depends on:

• Identifying common factors or patterns such as factoring by grouping or using special products like the difference of squares.
• Using the Rational Zero Theorem: This theorem helps in pinpointing potential rational zeros by examining factors of the constant term and the leading coefficient.

In our case, knowing the possible rational zeros from the factors \(\pm1, \pm2, \pm4\) aids in testing whether we can simplify \(f(x) = -x^3 + x^2 - 4x - 4\) further. Factorization is a powerful tool in a mathematician's arsenal for solving polynomial functions.