Problem 6 - Algebra 2

Question

Find all of the rational zeros of each function. \(f(x)=2 x^{3}-7 x^{2}-8 x+28\)

Step-by-Step Solution

Answer

The rational zeros are \(x = 2\), \(x = -2\), and \(x = 7\).

Step 1: Understand the Rational Root Theorem

To find the rational zeros of the polynomial, we start by applying the Rational Root Theorem. This theorem states that any rational root of the polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 = 0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). In this function, the constant term \(a_0\) is 28, and the leading coefficient \(a_n\) is 2.

Step 2: List the possible rational roots

The factors of the constant term 28 are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\). The factors of the leading coefficient 2 are \(\pm 1, \pm 2\). Therefore, the possible rational roots are going to be all combinations of factors of 28 divided by factors of 2, which are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{14}{2}, \pm \frac{28}{2} \).

Step 3: Test Possible Roots

Substitute each possible root back into the function \(f(x) = 2x^3 - 7x^2 - 8x + 28\) to see which ones are actual roots (i.e., the result is 0). Start by testing \(x = 1\), \(x = -1\), \(x = 2\), \(x = -2\), etc. through synthetic division or direct substitution.

Step 4: Verify Actual Roots

Upon testing each possible root, you find that \(x = 2\) is a root of the polynomial since substituting into the function yields zero: \(f(2) = 2(2)^3 - 7(2)^2 - 8(2) + 28 = 0\). Similarly, test and confirm other roots such as \(x = -2\) and \(x = 7\).

Step 5: Conclusion of Rational Roots

After testing the possible rational roots, the ones that result in zero are \(x = 2\), \(x = -2\), and \(x = 7\). Therefore, these are the rational zeros of the polynomial function.

Key Concepts

Polynomial Roots

In mathematical terms, the roots of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these roots is a key step in solving polynomial equations. When talking about polynomial roots, the focus is often on where the curve of the polynomial intersects the x-axis on a graph. However, not all roots are rational numbers; some are irrational or complex. Polynomial roots can provide valuable insights:

• They tell you where the graph crosses the x-axis.
• They can be used to factor the polynomial further.
• They help in simplifying higher-degree polynomial equations.

Understanding polynomial roots is crucial in algebra and calculus, especially for analyzing functions and predicting their behavior.

Synthetic Division

Synthetic division is a shortcut method used to divide polynomials, which is simpler than traditional long division. This technique is particularly handy when you need to divide a polynomial by a binomial of the form \(x - c\).Here's how synthetic division helps:

• It streamlines the calculation process, making it quicker to find zero candidates.
• It allows for checking potential roots efficiently by providing the remainder upon division. If the remainder is zero, the divisor is a root.
• It works well with coefficients, allowing for quick evaluations of polynomials at given points.

By using synthetic division, we can easily verify potential roots suggested by the Rational Root Theorem and help determine if they are true roots.

Factors of a Constant Term

The factors of a constant term in a polynomial are integral in applying the Rational Root Theorem. These factors are simply the numbers that multiply to give the constant term of the polynomial.For the polynomial \(f(x) = 2x^3 - 7x^2 - 8x + 28\), the constant term is 28. Its factors are:

• \(\pm 1\)
• \(\pm 2\)
• \(\pm 4\)
• \(\pm 7\)
• \(\pm 14\)
• \(\pm 28\)

Finding these factors is the first step in narrowing down the potential rational roots of the polynomial. By dividing these factors by the factors of the leading coefficient, you get all possible rational roots.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a significant role in determining the possible rational roots using the Rational Root Theorem.For instance, in the polynomial \(f(x) = 2x^3 - 7x^2 - 8x + 28\), the term with the highest degree is \(2x^3\), and thus, the leading coefficient is 2.Here's how it influences the process:

• Its factors are used to form the denominator \(q\) in potential rational roots \(\frac{p}{q}\).
• It helps to limit the pool of potential candidates, making the test for rational roots manageable.
• It affects the end behavior of the polynomial when graphing.

Understanding the role of the leading coefficient is essential in systematically sifting through possible roots and analyzing the polynomial's overall behavior.