Problem 37 - Precalculus with Calculus Previews

Question

In Problems 37 and 38 , find a polynomial function \(f\) of the indicated degree with integer coefficients that possesses the given rational zeros. $$ \text { degree } 4 ;-4, \frac{1}{3}, 1,3 $$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = 3x^4 - x^3 + x^2 + 49x - 12.\)

Step 1: Identify the roots

The given rational zeros of the polynomial are \[x = -4, \frac{1}{3}, 1, \text{ and } 3.\] This means that these are the roots of the polynomial function.

Step 2: Write the factor form of the polynomial

Since each root corresponds to a factor of the polynomial, write the polynomial in factor form based on the roots: \[(x + 4)(x - \frac{1}{3})(x - 1)(x - 3).\]

Step 3: Use integer coefficients

To ensure integer coefficients, multiply the factor \((x - \frac{1}{3})\) by 3 to eliminate the fraction. This results in the polynomial: \[3(x + 4)(x - \frac{1}{3})(x - 1)(x - 3) = (x + 4)(3x - 1)(x - 1)(x - 3).\]

Step 4: Expand the polynomial

To find the polynomial function, expand the expression by multiplying out the factors. First, multiply \[ (3x - 1)(x - 1) = 3x^2 - 3x - x + 1 = 3x^2 - 4x + 1.\] Then, multiply this result by the next remaining polynomial factor \[(x - 3)(3x^2 - 4x + 1) = (x + 4)(3x^3 - 13x^2 + 13x - 3).\]

Step 5: Continue expanding

Multiply the expression obtained in Step 4 by \((x + 4)\): \[(x + 4)(3x^3 - 13x^2 + 13x - 3) = x(3x^3 - 13x^2 + 13x - 3) + 4(3x^3 - 13x^2 + 13x - 3).\]

Step 6: Finalize the expression

Perform the multiplication and combine like terms for the final expanded polynomial:1. Multiply first:\[x(3x^3 - 13x^2 + 13x - 3) = 3x^4 - 13x^3 + 13x^2 - 3x\]2. Multiply second:\[4(3x^3 - 13x^2 + 13x - 3) = 12x^3 - 52x^2 + 52x - 12\]3. Add them together:\[3x^4 - x^3 + x^2 + 49x - 12\]

Step 7: Simplify

After combining all terms, the final polynomial function is: \[f(x) = 3x^4 - x^3 + x^2 + 49x - 12.\] This polynomial function has integer coefficients and all the given roots.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a fundamental concept when dealing with polynomial functions. It refers to the highest power of the variable within the polynomial expression. For instance, in our exercise, we aimed to find a polynomial of degree 4. This means that the highest power of the variable, typically denoted as \(x\), is 4 in the resulting polynomial expression.
The degree of a polynomial gives us crucial insight into the behavior of its graph and its possible number of roots. Specifically:

• The graph of a degree 4 polynomial can have up to 3 turning points.
• It can intersect the x-axis at most 4 times, as it can have up to 4 real roots.
• The end behavior of the graph is determined by the sign of the leading coefficient.

Understanding the degree helps in anticipating the overall structure of polynomial graphs and plays a vital role in predicting how the polynomial behaves across its domain.

Integer Coefficients

Working with integer coefficients in polynomials ensures that the polynomial's terms are simple and free from irrational or fractional components. In mathematical problems such as the one we tackled, it is crucial that all coefficients, the numbers multiplying the variables, be integers.
Integer coefficients make computations more reliable and the polynomial easier to communicate and interpret. Here are some reasons why integer coefficients are important:

• They maintain the simplicity and clarity of a polynomial function.
• They are generally easier for manual calculations and manipulations.
• Many applications in algebra and number theory require integer coefficients for solvability or further implications.

In our problem, to convert the rational zero \(\frac{1}{3}\) to a root with integer coefficients, we multiplied the corresponding factor by 3. This adjustment ensures all coefficients in the expanded polynomial are integers. The final polynomial was verified to maintain integer coefficients across all terms, fulfilling one of the key requirements of the problem.

Rational Zeros

Rational zeros of a polynomial refer to the solutions of the polynomial equation that can be expressed as fractions of integers or whole numbers. When given rational zeros, like in our exercise, it is essential to translate these into factors of the polynomial.
The Rational Root Theorem is a tool that helps in identifying these zeros. Specifically, rational zeros can be derived from the factors of the constant term and the leading coefficient. Here's a bit more detail on the process:

• Each given rational zero \(\frac{a}{b}\) translates directly to a factor \((bx - a)\).
• This approach ensures all roots are correctly incorporated into the polynomial.
• Multiplying out these factors as in our expansion ensures a complete polynomial is obtained.

In the exercise, the rational zeros \(-4, \frac{1}{3}, 1, \text{ and } 3\) guided the formation of polynomial factors. By multiplying these factors correctly, especially adjusting for integer coefficients, the resulting polynomial meets all given conditions and accurately represents the provided rational zeros.