Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(1-\sqrt{3}, 1+\sqrt{3},\) and 1

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^3 - 3x^2 + 2\).

Step 1: Identify Given Zeros

Note that the polynomial has the given zeros: \(1-\sqrt{3}, 1+\sqrt{3},\) and \(1\). This implies the polynomial takes the form \((x - (1-\sqrt{3}))(x - (1+\sqrt{3}))(x - 1)\).

Step 2: Convert Roots into Factors

Convert each zero into factors of the polynomial. Use the expression highlighting subtraction from \(x\) for each zero, which gives factors of the form \((x - 1 + \sqrt{3})(x - 1 - \sqrt{3})(x - 1)\).

Step 3: Simplify Polynomial Expression

Multiply the factors \((x - 1 + \sqrt{3})\) and \((x - 1 - \sqrt{3})\) using the difference of squares formula: \[ (x - 1)^2 - (\sqrt{3})^2 = (x - 1)^2 - 3 \]This yields \((x - 1)^2 - 3 = x^2 - 2x + 1 - 3 = x^2 - 2x - 2\).

Step 4: Multiply Remaining Factor

Now multiply this result by the remaining factor \((x - 1)\):\[ (x^2 - 2x - 2)(x - 1) = x^3 - x^2 - 2x - x^2 + 2x + 2 = x^3 - 3x^2 \] Thus, we acquire the polynomial \(P(x) = x^3 - 3x^2 + 2\).

Step 5: Verify Degree and Coefficients

Check that the degree of the polynomial is 3, which is the least possible degree for three zeros. Confirm that the leading coefficient is 1 and that all coefficients are real.

Key Concepts

Roots and zeros of polynomials

Understanding roots and zeros of a polynomial is critical when analyzing or constructing these functions.

A **root** or **zero** of a polynomial is a value of the variable (usually denoted as \(x\)) that makes the polynomial equal to zero. In mathematical terms, if \(P(x)\) is a polynomial, then \(c\) is a root if \(P(c) = 0\).

For example, given zeros like \(1-\sqrt{3}, 1+\sqrt{3},\) and \(1\), the polynomial must yield zero when any of these values are substituted into it.

Understanding this concept helps in forming the polynomial equation, as it can be broken down into linear factors derived from its zeros, such as \((x - (1-\sqrt{3}))(x - (1+\sqrt{3}))(x - 1)\). Each factor represents a root.

Remember, the number of roots of a polynomial corresponds to its degree, assuming all roots are counted regarding their multiplicity.

Degree of polynomials

The **degree** of a polynomial is one of its most fundamental characteristics. It is defined as the highest power of the variable \(x\) that appears in the polynomial.

For instance, the polynomial \(P(x) = x^3 - 3x^2 + 2\) has a degree of 3 because the variable \(x\) is raised to the power of 3 in its highest term.

• Higher-degree polynomials can have more complex roots.
• The degree of the polynomial indicates how many roots (or zeros) it can have.
• In this context, the degree of a polynomial is crucial, as it directly relates to the polynomial's ability to have zeros \((1-\sqrt{3}, 1+\sqrt{3},\) and \(1\)).

Checking that the polynomial formed has the least possible degree means ensuring no root is repeated unnecessarily, keeping the polynomial simpler and more efficient.

Difference of squares

The **difference of squares** is a specific formula used for simplifying expressions, which appears in many areas of mathematics, especially polynomial operations.

It is expressed as:

1. \(a^2 - b^2 = (a + b)(a - b)\)

This straightforward identity helps rapidly simplify expressions involving squares.

In our polynomial, we apply it during the simplification of the expression \((x - 1 + \sqrt{3})(x - 1 - \sqrt{3})\), yielding

1. \((x - 1)^2 - (\sqrt{3})^2 = (x - 1)^2 - 3\)
2. Which simplifies further to \(x^2 - 2x - 2\).

Utilizing the difference of squares is efficient because it reduces multiplicative complexity, leading to faster and easier calculation.

Real coefficients in polynomials

Real coefficients in a polynomial mean all the numbers before the variables and constant terms are real numbers. This concept is vital as it influences the behavior and graph of the polynomial.

Real coefficients imply that the polynomial can interact smoothly with real-world applications, where complex numbers are often less meaningful. In this exercise, forming a polynomial with given zeros \(1-\sqrt{3}, 1+\sqrt{3},\) and \(1\) while ensuring real coefficients involves ensuring each coefficient (like -3 in \(x^3 - 3x^2 + 2\)) remains a real number.

• It enhances the practical usability of the polynomial.
• Keeps calculations straightforward and relatable.
• Ensures easier interpretation of the polynomial's graph.

Polynomial equations often need to have real coefficients, especially in situations modeled or understood in real numbers, enhancing clarity and application.