Problem 1 - Precalculus with Calculus Previews

Question

In Problems \(1-20,\) find all rational zeros of the given polynomial function \(f\). $$ f(x)=5 x^{3}-3 x^{2}+8 x+4 $$

Step-by-Step Solution

Answer

The polynomial has no rational zeros.

Step 1: Identify Possible Rational Zeros

To find all rational zeros of the polynomial \( f(x) = 5x^3 - 3x^2 + 8x + 4 \), we use the Rational Root Theorem. This theorem tells us that any rational zero, \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term (4) and \( q \) is a factor of the leading coefficient (5). So, possible values for \( p \) are ±1, ±2, ±4, and possible values for \( q \) are ±1, ±5. This gives us potential rational zeros: ±1, ±2, ±4, ±\( \frac{1}{5} \), ±\( \frac{2}{5} \), ±\( \frac{4}{5} \).

Step 2: Test Possible Rational Zeros

Evaluate \( f(x) \) for each possible rational zero. This involves substituting the values into the polynomial until a zero is found. For instance, start with \( f(1) = 5(1)^3 - 3(1)^2 + 8(1) + 4 = 14 \), which is not zero. Continue this process for other candidates, such as \( f(-1) \), \( f(2) \), and so on until a rational zero is found.

Step 3: Verification Through Factorization

Suppose you find, for example, \( x = -1 \) to be a zero (it actually isn't, based on our initial test), then \( x + 1 \) is a factor of \( f(x) \). Perform polynomial division of \( f(x) \) by \( x + 1 \) to reduce the polynomial order and re-evaluate for further zeros. This division is important to check if any other rational zeros exist in the reduced polynomial.

Step 4: Analytical Checking for Errors

Since no rational zeros were found in previous steps upon verification, double-check each test and ensure correct substitutions into the polynomial. All candidates must follow through the defined outcome ensuring no computational errors in aligned outcomes (calculated values differing from zero).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can conveniently describe a wide range of curves and geometric shapes. Polynomials are often used in algebra and calculus to solve various kinds of problems. In the given problem, the polynomial function is represented as:\[ f(x) = 5x^3 - 3x^2 + 8x + 4 \]This function is a cubic polynomial where 5 is the leading coefficient (the coefficient of the term with the highest power), and 4 is the constant term. Each term in the polynomial has an integer coefficient, and the variable "x" is raised to whole number powers.

Rational Zeros

The Rational Root Theorem is a useful tool for finding the rational zeros of a polynomial function. A rational zero is a solution of the form \( \frac{p}{q} \) where both \( p \) and \( q \) are integers.

• According to the theorem, \( p \) is a factor of the constant term of the polynomial, which is 4 in this case.
• And, \( q \) is a factor of the leading coefficient, which is 5 here.

Through the Rational Root Theorem, the possible rational zeros for our polynomial becomes:\[ \pm 1, \pm 2, \pm 4, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5} \] Testing these potential zeros systematically checks for any real zeros that can satisfy the polynomial equation.

Polynomial Division

Polynomial division is used when you find a rational zero of a polynomial. The technique helps decrease the degree of the polynomial, making it simpler to find further zeros.To perform polynomial division, place the polynomial you are dividing by in front. Divide the terms starting from the highest degree downwards. For instance, if \( x = a \) is a zero of the polynomial \( f(x) \), then \( x - a \) is a factor. Using polynomial division:

• If \( f(a) = 0 \), then divide \( f(x) \) by \( x-a \) to get a quotient.
• This reduces the degree of the polynomial and helps reveal other zeros.

Be sure to perform each division step correctly to ensure you identify all possible rational zeros.

Factorization

Factorization involves expressing the polynomial as a product of its factors. If you identify a zero of the polynomial, you can use it to factor the polynomial further. As an example:If \( x = a \) is a known zero, then the polynomial is divisible by \( x-a \).By factoring the polynomial, you divide the original polynomial by its known factors.

• This reduces the polynomial to a simpler form.
• Understanding the factors helps to identify and confirm all zeros of the polynomial.

Starting from the polynomial \( f(x) = 5x^3 - 3x^2 + 8x + 4 \), if you successfully find a factorization, each factor gives additional insights into further solving the polynomial equation. This step is key in confirming and understanding the complete roots of the polynomial, both real and complex.