Problem 19 - 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. 5 and \(-4\)

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^2 - x - 20\).

Step 1: Write the Factors

Since the given zeros are 5 and -4, the corresponding factors of the polynomial are \((x - 5)\) and \((x + 4)\). Each zero corresponds to setting the polynomial equal to zero, i.e., solving \(x - 5 = 0\) gives the zero 5, and \(x + 4 = 0\) gives the zero -4.

Step 2: Multiply the Factors

To find the polynomial, multiply the factors obtained in Step 1: \((x - 5)(x + 4)\). This multiplication will give the polynomial with the specified zeros.

Step 3: Expand the Expression

Expand \((x - 5)(x + 4)\) using the distributive property:\[(x - 5)(x + 4) = x(x + 4) + (-5)(x + 4)\]First: \(x \cdot x = x^2\), \(x \cdot 4 = 4x\)Second: \(-5 \cdot x = -5x\), \(-5 \cdot 4 = -20\)Combine these results: \(x^2 + 4x - 5x - 20\).

Step 4: Simplify the Expression

Combine like terms in the expanded expression:\(x^2 + 4x - 5x - 20 = x^2 - x - 20\).This polynomial has a leading coefficient of 1, and degree 2, which satisfies the condition of having real coefficients and the least possible degree.

Key Concepts

Zeros of Polynomial

The concept of "Zeros of a Polynomial" is fundamental when dealing with polynomial functions. Let's delve into what it means. In simple terms, the zeros of a polynomial are the values of the variable (often represented as \(x\)) that make the polynomial equal to zero. Practically, these zeros are the solutions to the polynomial equation \(P(x) = 0\). When you have a polynomial like \(x^2 - x - 20\), the zeros are the values of \(x\) that satisfy this equation.
When given a polynomial function \(P(x)\), finding its zeros involves solving the equation created by setting the polynomial to zero. In our exercise, the polynomial was formed from given zeros, namely 5 and -4. These zeros imply that when \(x = 5\) and \(x = -4\), the polynomial evaluates to zero. Therefore, the task often begins with identifying these values through understanding the polynomial's factors.
This process is crucial for graphing the polynomial, as each zero corresponds to an \(x\)-intercept on its graph. Moreover, identifying the zeros helps in analyzing the polynomial's behavior and properties.

Factoring Polynomials

Factoring polynomials is a key method to uncover the zeros of a polynomial function and simplify polynomial expressions. The goal of factoring is to rewrite the polynomial as a product of simpler polynomials, often linear factors like \((x - a)\), where \(a\) is a zero of the polynomial.
In the original exercise, given zeros 5 and -4, we formed factors \((x - 5)\) and \((x + 4)\). These factors correlate with the zeros because they reflect the values for \(x\) that make the polynomial zero. Essentially, the process of factoring reverses the expansion of a polynomial, breaking it down into its elementary components.

• Recognizing common factors in terms,
• Arranging terms to use special formulas or identities,
• Applying the quadratic formula if necessary.

Once a polynomial is factored, solving the polynomial equation becomes straightforward—set each factor to zero and solve for \(x\). This reveals the polynomial's zeros, critical for solving equations and graphing the functions.

Distributive Property

The distributive property is a fundamental algebraic rule used extensively in the expansion and simplification of polynomials. It allows us to distribute multiplication over addition or subtraction in an expression, making it simpler to manage and understand.
Consider the exercise where you had to expand \((x - 5)(x + 4)\). Using the distributive property, each term in the first binomial \((x - 5)\) is multiplied by each term in the second binomial \((x + 4)\). This yields:

• First, \(x \cdot x = x^2\)
• Then, \(x \cdot 4 = 4x\)
• Next, \(-5 \cdot x = -5x\)
• Finally, \(-5 \cdot 4 = -20\)

By combining these products, you obtain the expanded polynomial \(x^2 + 4x - 5x - 20\). Subsequently, you can simplify by combining like terms, resulting in \(x^2 - x - 20\). The distributive property is indispensable because it facilitates the expansion of products into polynomials, essential in deriving and analyzing polynomial equations.