Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=x^{3}-4 x^{2}-4 x+16$$

Step-by-Step Solution

Answer

After applying the Rational Zero Test, the initial set of potential rational zeros were \(\pm1, \pm2, \pm4, \pm8, \pm16\), but only 2 is an actual rational zero of the polynomial \(f(x)=x^{3}-4x^{2}-4x+16\).

Step 1: Identification

The polynomial given here is \(f(x)=x^{3}-4x^{2}-4x+16\). The leading coefficient is 1 and the constant term is 16.

Step 2: List all factors

We list all factors of the leading coefficient (1) which are just \(\pm1\), and all factors of the constant term (16). The factors of 16 are \(\pm1\), \(\pm2\), \(\pm4\), \(\pm8\), and \(\pm16\).

Step 3: Form all possible rational zero

Make a list of all possible rational zeros as \(\frac{p}{q}\) where 'p' is a factor of 16 and 'q' is a factor of 1. This leads to the possible rational zeros: \(\pm1\), \(\pm2\), \(\pm4\), \(\pm8\), \(\pm16\).

Step 4: Find Actual Zeros

Substitute the possible rational zeros into the function \(f(x)\). The values which make \(f(x)=0\) are the actual rational zeros of the function. Substituting values, we see that \(f(2)=2^{3}-4\times2^{2}-4\times2+16 = 0\). Thus, 2 is a rational zero of the function. Other values do not make the equation equal to zero.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of the variable, each multiplied by a coefficient. They can be as simple as a constant, which is technically a polynomial of degree zero, or as complex as a sum of many different powers of the variable. In our exercise, we are dealing with a cubic polynomial: \[f(x)=x^{3}-4x^{2}-4x+16\] This polynomial has three terms with the variable raised to powers of 3, 2, and 1, plus a constant term. The degree of this polynomial is the highest power of the variable, in this case, 3. Polynomial functions are fundamental in mathematics as they are used to model various real-world phenomena.

• The degree tells us the maximum number of solutions (or zeros) the function may have.
• The coefficients can affect the shape and position of the graph of the polynomial.
• Each zero of the polynomial affects where the graph intersects the x-axis.

Understanding the structure of polynomial functions is crucial to finding their zeros efficiently.

Exploring Possible Rational Zeros

Finding the zeros of a polynomial function starts with determining the possible options. The Rational Zero Theorem is helpful because it allows us to list possible rational zeros without guessing. Given a polynomial function: \[f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\] The possible rational zeros are the fractions \(\frac{p}{q}\), where:

• '\(p\)' are the factors of the constant term \(a_0\).
• '\(q\)' are the factors of the leading coefficient \(a_n\).

In our case, for the polynomial \(f(x) = x^{3}-4x^{2}-4x+16\), the constant term is 16 and the leading coefficient is 1. Thus, possible values for \(p\) are the factors of 16 which include \(\pm1, \pm2, \pm4, \pm8, \pm16\). Since 1 is the only factor of the leading coefficient, \(q\) is \(\pm1\). That means our potential rational zeros are simply \(\pm1, \pm2, \pm4, \pm8, \pm16\), because dividing by ±1 leaves the factors unchanged.

Applying the Rational Zero Test

The Rational Zero Test helps in systematically verifying which of the possible rational zeros listed are actual zeros of the polynomial. Here's how you apply the test:

• Take each possible rational zero and substitute it into the polynomial.
• Simplify the expression. If the result is zero, then that number is indeed a zero of the polynomial.
• Repeat for all possible rational zeros.

For our polynomial \(f(x)=x^{3}-4x^{2}-4x+16\), we try each potential zero in turn. Specifically, when we insert \(2\) into the polynomial: \[f(2)=2^{3}-4 \times 2^{2} -4 \times 2 +16 = 0\] Since the result is zero, \(x=2\) is confirmed as a rational zero. Try others like \(\pm1, \pm4, \pm8, \pm16\), and you'll see they don't satisfy the zero condition. Therefore, they are not zeros of the polynomial.

Finding Zeros of a Polynomial

Finding zeros of a polynomial entails identifying the x-values where the polynomial equals zero. These zeros reveal where the graph of the polynomial intersects the x-axis. Here's the general approach:

• Use techniques like the Rational Zero Test to list potential zeros.
• Test these candidates to confirm which are true zeros.
• Once confirmed, you can factor the polynomial to find other non-rational zeros.

For our polynomial \(f(x)=x^{3}-4x^{2}-4x+16\), we discovered \(x=2\) as a rational zero. Factoring based on this zero, using synthetic or long division, can help uncover other zeros—potentially irrational or complex. Finding these zeros is crucial because they provide valuable insights into the behavior of the polynomial function. When combined with techniques like graphing, understanding polynomial zeros empowers one to solve complex equations and apply mathematical concepts to real-world situations.