Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function that has the given zeros. (There are many correct answers.) \(4+\sqrt{3}, 4-\sqrt{3}\)

Step-by-Step Solution

Answer

The polynomial function derived from the given zeros \(4 + \sqrt{3}\) and \(4 - \sqrt{3}\) is \(P(x) = x^2 - 8x + 19\).

Step 1: Understand the given zeros

The exercise provides two zeros for a polynomial function: \(4 + \sqrt{3}\) and \(4 - \sqrt{3}\). These zeros will be used to derive the polynomial function.

Step 2: Set up the equation

The polynomial function can be obtained by the product of binomial equations of the form \((x - a)\), where 'a' represents a zero of the function. Therefore, this step involves constructing these binomial equations using the zeros provided i.e., \((x - (4 + \sqrt{3}))\) and \((x - (4 - \sqrt{3}))\).

Step 3: Product of the binomials

Multiply the binomials derived in step2: \((x - (4 + \sqrt{3})) * (x - (4 - \sqrt{3}))\). Through applying the FOIL method, the polynomial function 'P(x)' is derived.

Step 4: Simplifying the expression

Simplify the expression from Step 3 by combining like terms and simplifying expressions with the variable 'x' and constant terms to obtain the final polynomial function.

Key Concepts

Zeros of Polynomials

Every polynomial function has zeros, or solutions, where the function's value equals zero. The zeros are critical because they define where the graph of the polynomial will intersect the x-axis. By knowing the zeros of a polynomial, you can work backwards to determine the original function. For instance, if the zeros are given as \(4+\sqrt{3}\) and \(4-\sqrt{3}\), it confirms that these points are where the polynomial will touch or cross the x-axis. For a polynomial equation \(P(x) = 0\), each zero corresponds to an \(x\)-value that makes \(P(x)\) equal to zero. Understanding how to derive a polynomial from its zeros is one of the foundational skills in algebra. Learning this helps in visualizing the function's behavior and analyzing its graphical representation.

Binomial Equations

To find the polynomial function from its zeros, you construct binomial equations based on those zeros. A binomial equation is typically expressed in the form of \((x-a)\), where 'a' is the zero of the polynomial. In the given problem, we want

• \(x-(4+\sqrt{3})\)
• \(x-(4-\sqrt{3})\)

These binomials are the building blocks to create our polynomial. Essentially, each binomial represents a linear factor of the polynomial, and multiplying these factors together gives the complete polynomial. Binomial equations help to visualize each step in forming the polynomial, ensuring that the derived polynomial reflects these critical zeros accurately.

FOIL Method

The FOIL method is a process used to multiply two binomials. It's an acronym that stands for First, Outer, Inner, and Last, referring to the terms you need to multiply in each step:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

Using the FOIL method on \((x-(4+\sqrt{3}))(x-(4-\sqrt{3}))\) results in expanding and then simplifying these products:

• First: \(x \times x = x^2\)
• Outer: \(x \times -(4-\sqrt{3})\)
• Inner: \(-(4+\sqrt{3}) \times x\)
• Last: \(-(4+\sqrt{3})(4-\sqrt{3})\)

Employing the FOIL method ensures that each term is considered, leading to a full expansion of the polynomial product.

Simplifying Polynomials

Once we expand the polynomial using the FOIL method, the expression often involves several terms which need simplification. Simplifying polynomials means combining like terms and reducing the equation to its simplest form.

• After expanding with FOIL, we calculate and combine terms that contain the same degree of \(x\).
• Constant terms are also simplified; for example, in our equation: \( (4 + \sqrt{3})(4 - \sqrt{3}) = 16 - 3 = 13\).

This step is crucial because it reduces the polynomial to its cleanest form, making it easier to analyze and interpret. A simplified polynomial presents clearly defined coefficients and terms, standalone from unnecessary complexity. This step not only helps in clarifying the function but also prepares it for further algebraic operations or graphing.