Problem 9 - 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. $$ f(x)-x^{3}+x^{2}-4 x-4 $$

Step-by-Step Solution

Answer

The rational zeros of the function \(f(x)=x^{3} + x^{2} - 4x - 4\) are -1, 2, and -2.

Step 1: Listing All Possible Rational Zeros

According to the Rational Root Theorem, the rational roots of a polynomial are determined by the quotient of the factors of the constant term divided by the factors of the leading coefficient. For our problem, the constant is -4 and the leading coefficient is 1. The factors of -4 are -1, 1, -2, 2, -4, 4 and those of 1 are 1 and -1. So, the possible rational zeros of the function are 1, -1, 2, -2, 4, -4.

Step 2: Synthetic Division to Find an Actual Zero

Using synthetic division, we test each possible root until we find one that yields 0 as a remainder. In our case, -1, will yield 0 as remainder, thus -1 is an actual root of the polynomial.

Step 3: Use Quotient From Synthetic Division to Find Remaining Zeros

The quotient we get from synthetic division when an actual zero is found will be a polynomial of lower degree. For our problem, the quotient when dividing by x+1, is \(x^2 - 4\) equating the quotient to zero and solving the equation will yield the remaining roots. This gives us \(x = 2\) and \(x = -2\) , which are the remaining zeros of the polynomial function.

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, featuring operations of addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial function is:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients, and they can include zero values.
• The highest power of the variable, \(n\), determines the degree of the polynomial.

Understanding polynomial functions is essential as they form the groundwork for more complex mathematical concepts. In our exercise, the polynomial function \(f(x) = x^3 + x^2 - 4x - 4\) is a third-degree polynomial due to its highest power, 3. The solutions (or zeros) of this polynomial are the values of \(x\) that make \(f(x) = 0\). These zeros are critical in understanding the behavior and graph of the function.

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear divisor of the form \((x - c)\). It's particularly useful in finding zeros of polynomials, as it allows for quick computation and verification without lengthy calculations.To perform synthetic division:

• Write down the coefficients of the polynomial you are dividing.
• Write the zero of the divisor, which is \(c\) from \((x-c)\).
• Bring down the leading coefficient.
• Multiply the zero by the number just written and write it beneath the next coefficient.
• Add the column and continue until completion.

Using synthetic division with our polynomial \(f(x)=x^3+x^2-4x-4\) and testing each possible rational zero, we found that \(-1\) is an actual zero since it caused the remainder to be zero. The process gives us a new reduced equation or quotient, which helps in discovering the remaining zeros.

Rational Roots

The Rational Root Theorem is a useful tool for finding the rational zeros (roots) of a polynomial function. It states that if a polynomial has a rational zero \(\frac{p}{q}\), then \(p\) (a factor of the constant term \(a_0\)) and \(q\) (a factor of the leading coefficient \(a_n\)) are integers.Here's how you apply the theorem:

• List all factors of the constant term \(a_0\).
• List all factors of the leading coefficient \(a_n\).
• Express all potential rational roots as \(\frac{p}{q}\), ensuring each \(p\) comes from the factors of the constant and each \(q\) from the factors of the leading coefficient.

For \(f(x)=x^3+x^2-4x-4\), our rational root possibilities are \(\pm1, \pm2, \pm4\). Testing these through synthetic division aids in pinpointing actual zeros. Once we find actual zeros, like \(-1\) in our case, other methods or synthetic division itself can be used for further analysis to find remaining roots.