Problem 19 - A Graphical Approach to College Algebra

Question

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$5 \text { and } -4$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Zeros

The problem provides two zeros for the polynomial function, which are 5 and -4.

Step 2: Form Linear Factors

From the zeros provided, we can form the linear factors \((x - 5)\) corresponding to the zero 5, and \((x + 4)\) corresponding to the zero -4.

Step 3: Write the Polynomial Function

Multiply the linear factors to form the polynomial function: \[(x - 5)(x + 4)\]

Step 4: Expand the Expression

Expand the expression obtained from multiplying the linear factors:\[(x - 5)(x + 4) = x^2 + 4x - 5x - 20\] Combine like terms:\[x^2 - x - 20\]

Step 5: Verify the Polynomial

Check that the polynomial has a leading coefficient of 1, the least possible degree (2 since there are two roots), and real coefficients. Our polynomial, \[x^2 - x - 20\] meets all these criteria.

Key Concepts

Zeros of Polynomials

In the realm of polynomial functions, the zeros of a polynomial are critical to understand. A zero of a polynomial function is a value of the variable that makes the function equal zero. In simpler terms, if you plug a zero into the polynomial, the result will be zero. For example, if a polynomial function is given by \(P(x) = x^2 - x - 20\), and we know its zeros are 5 and -4, then \(P(5) = 0\) and \(P(-4) = 0\).
Finding the zeros is crucial because it allows us to express the polynomial in its factorized form, which can simplify solving equations or analyzing the graph of the function. Remember, the zeros can be real numbers like in this exercise, or they might sometimes be complex numbers depending on the polynomial's nature.
Understanding zeros provides deep insights into the behavior and characteristics of polynomial functions, especially when combined with other concepts like linear factors.

Linear Factors

Linear factors are an essential concept when dealing with polynomial functions, especially when their zeros are known. Given zeros help us form these linear factors. For each zero of the polynomial, there corresponds a linear factor. For instance, if a polynomial has zeros at 5 and -4, the linear factors will be \((x - 5)\) and \((x + 4)\).

• Each linear factor is derived by subtracting the zero from \(x\). If the zero is \(5\), the factor becomes \((x - 5)\).
• The beauty of linear factors is that when they are multiplied together, they produce the original polynomial.

In this case, the linear factors \((x - 5)\) and \((x + 4)\) multiply to give our polynomial \((x - 5)(x + 4) = x^2 - x - 20\).
Working with linear factors is not just limited to creating polynomials; it also plays a significant role in simplifying complex polynomial expressions and solving polynomial equations efficiently.

Real Coefficients

Polynomial functions often have coefficients, which are the numerical factors accompanying the variable terms. When we say a polynomial has real coefficients, it means all these accompanying numbers are real numbers, not complex or imaginary. Real coefficients are important because they ensure that the polynomial will behave predictably within the real number system.

• Having real coefficients is crucial for representing phenomena in the real world, where imaginary numbers do not apply.
• They guarantee that any real zeros of the polynomial will be accurate and reliable for applications.

In our exercise, where the polynomial \(x^2 - x - 20\), its coefficients (1, -1, and -20) are indeed real numbers. This property is often a requirement when constructing polynomial functions, especially in real-world situations where values need to make tangible sense.
Establishing that coefficients are real ensures clarity and validity in mathematical operations involving polynomial functions.

Degree of Polynomial

The degree of a polynomial is a key concept that indicates the highest power of the variable in the polynomial expression. It is reflective of the polynomial's complexity and the number of possible roots or zeros it can have. For instance, a polynomial like \(x^2 - x - 20\) is of degree 2 because the highest exponent of \(x\) is 2.

• Polynomials with degree 2 are known as quadratics, and they can have at most two zeros.
• The degree often sets the stage for the number of inflection points and the general shape of the graph.

Higher-degree polynomials can be more complex, with more potential changes in direction on their graphs. In our exercise, achieving the least degree possible (which is 2 in this example) was necessary to keep the polynomial simple while meeting criteria like maintaining a leading coefficient of 1.
Understanding the degree of a polynomial helps in graphing and solving equations, as it provides a roadmap for the function's fundamental characteristics.