Problem 15 - College Algebra

Question

In Exercises \(9-16\) a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quotient from part (b) to find the remaining zeros of the polynomial function. $$ f(x)-2 x^{3}+6 x^{2}+5 x+2 $$

Step-by-Step Solution

Answer

The zeros of the polynomial \(f(x) = 2x^3 + 6x^2 + 5x + 2\) are \(x = -2, -1, 1\).

Step 1: Listing All Possible Rational Zeros Using The Rational Root Theorem

The rational root theorem states that any possible rational zero of a polynomial can be expressed as a fraction \( \frac{p}{q} \), where p is a factor of the constant term (2 in this case) and q is a factor of the leading coefficient (2 in this case as well). Thus, for the polynomial \(2x^3 + 6x^2 + 5x + 2\), the possible rational zeros are \(\pm1, \pm2\) which is calculated by \(\pm \frac{p}{q}\).

Step 2: Using Synthetic Division to Test the Possible Rational Zeros and Find an Actual Zero

Start synthetic division process using 1, because it's the smallest positive integer and see whether this yields a remainder of zero. You setup the synthetic division, where 1 is placed outside of the bracket, and the coefficients of the polynomial are inside. If after you perform synthetic division the remainder is zero, 1 is a root of the polynomial. Synthetic division with root 1: \[\begin{array}{c|cccc} 1 & 2 & 6 & 5 & 2 \ \ & & 2 & 8 & 13 \ \ \hline & 2 & 8 & 13 & 15\end{array}\] The remainder is 15, not 0, so 1 is not a root. Try -1 as a root. Setting up synthetic division like before, and if the remainder is zero, -1 is a root of the polynomial. Synthetic division with root -1: \[\begin{array}{c|cccc} -1 & 2 & 6 & 5 & 2 \ \ & & -2 & -4 & -1 \ \ \hline & 2 & 4 & 1 & 1\end{array}\] The remainder is 1, not 0, so -1 is not a root either. Proceed with this method until finding a root. Synthetic division with root 2: \[\begin{array}{c|cccc} 2 & 2 & 6 & 5 & 2 \ \ & & 4 & 20 & 50 \ \ \hline & 2 & 10 & 25 & 52\end{array}\] The remainder is 52, not 0, so 2 is not a root. Finally try -2 as a root. Synthetic division with root -2: \[\begin{array}{c|cccc} -2 & 2 & 6 & 5 & 2 \ \ & & -4 & 4 & -2 \ \ \hline & 2 & 2 & 9 & 0\end{array}\] The remainder is 0, so -2 is a root of the polynomial function.

Step 3: Using the Quotient to Find the Remaining Zeros

The solutions to the polynomial can be found by setting it to zero and solving for x: \[-2x^3+2x + 9 = 0\]. This simplifies to: \[-2x(x^2 -1) + 9 = 0 \]. This gives two additional solutions: \[x = 1, -1\].

Key Concepts

Synthetic Division

Synthetic division is a simple and efficient method used to divide polynomials, especially useful when dividing by a linear factor of the form \(x - c\). This technique significantly reduces the complexity compared to traditional polynomial long division.

Here's how synthetic division works:

• First, identify the polynomial you are working with and the potential root \(c\) you are testing.
• Write down the coefficients of the polynomial in descending order of power.
• Place the potential root \(c\) to the left side (outside) of a vertical bar, and the coefficients to the right.
• Pull down the first coefficient as it is.
• Multiply this number by \(c\) and write the result under the next coefficient.
• Add vertically and place the result directly below.
• Continue this process until you've worked through all coefficients.

If the last number (remainder) is zero, \(c\) is a root of the polynomial. Otherwise, it is not a root, and you must try another potential root.
This method helps verify potential rational zeros by quickly identifying whether they are actual roots without extensive calculations.

Polynomial Roots

Polynomial roots, also known as zeros, are the values of \(x\) where the polynomial function equals zero. For a polynomial function \(f(x) = 0\), roots are the solutions to this equation. These could be real or complex numbers, depending on the polynomial and its degree.

Determining the roots of a polynomial is essential in understanding the behavior of the function. Here are some key points to remember:

• The degree of the polynomial indicates the maximum number of possible roots.
• Roots can be either real, complex, or repeated.
• Finding one root sometimes helps in reducing the polynomial’s degree, making it easier to find other roots.

To find the roots, techniques such as synthetic division, factoring, the use of the quadratic formula, or graphing are often employed. Synthetic division is particularly useful when working with polynomials to simplify equations and find real roots efficiently.

Rational Zeros

Rational zeros are the values of \(x\) that make the polynomial function equal to zero and can be expressed as fractions. The Rational Root Theorem helps to list all possible rational zeros of a given polynomial, making it easier to test and find actual rational roots.

The Rational Root Theorem states:

• Any rational solution, \(\frac{p}{q}\), of the polynomial is such that \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.

This theorem significantly narrows down the possible rational numbers that need to be tested. Once these potential rational zeros are listed, synthetic division can be used to determine if any of these potential zeros are actual zeros.
Testing potential rational zeros can quickly identify the actual roots, simplifying complex polynomials and allowing for further factoring or solving.

College Algebra

College Algebra is a course that covers fundamental concepts of algebra at a more advanced level than high school algebra. The course often includes a variety of topics such as polynomial functions, rational expressions, exponential and logarithmic functions, systems of equations, and inequalities.

Understanding concepts such as polynomial roots, rational zeros, and synthetic division is crucial for success in College Algebra. These topics form the foundation for more advanced studies in mathematics and are often utilized in various fields that require analytical skills.

Key components of College Algebra include:

• Comprehending and applying algebraic expressions and functions.
• Solving complex equations and inequalities.
• Utilizing mathematical reasoning to solve real-life problems.
• Developing skills to interpret and model mathematical situations.

By mastering these topics, students are well-prepared for courses in calculus and other higher-level mathematics, essential for careers in engineering, physics, economics, and more.