Problem 33 - Algebra 2

Question

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(-4,1,5\)

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x^3 - 2x^2 - 19x + 20\).

Step 1: Understand the Relationship Between Zeros and Factors

When you are given zeros of a polynomial function, such as \(-4, 1,\) and \(5\), each zero corresponds to a factor of the polynomial. Thus, if \(x = a\) is a zero, then \(x - a\) is a factor. Therefore, the factors related to the given zeros are \((x + 4), (x - 1), \text{and}\ (x - 5)\).

Step 2: Write the Polynomial as a Product of Factors

Now write the polynomial function as a product of its factors. Since the zeros of the polynomial are \(-4, 1, \text{and } 5\), the polynomial can be written as \(f(x) = (x + 4)(x - 1)(x - 5)\).

Step 3: Expand the Factored Form

To find the polynomial in standard form, expand \((x + 4)(x - 1)(x - 5)\):1. First, multiply the first two factors: \[(x + 4)(x - 1) = x^2 - x + 4x - 4 = x^2 + 3x - 4\]2. Next, multiply the result with the third factor: \[ (x^2 + 3x - 4)(x - 5)\] Distribute each term: \[x^3 - 5x^2 + 3x^2 - 15x - 4x + 20\] Combine like terms: \[x^3 - 2x^2 - 19x + 20\].

Step 4: Finalize the Polynomial Function

The polynomial function of least degree with integral coefficients having zeros at \(-4, 1,\) and \(5\) is:\[f(x) = x^3 - 2x^2 - 19x + 20\]

Key Concepts

Zeros of a Polynomial

Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. These values, denoted as roots or solutions, are crucial because they provide important insights about the behavior of the polynomial function. For instance, if you know the zeros of a polynomial, you can reconstruct the polynomial itself.

For example, if a polynomial has zeros at

• derivative derivations,-4
• 1
• 5

this means that plugging in these values into the polynomial function will yield a result of zero. Mathematically, this can be represented as

• f(-4) = 0
• f(1) = 0
• f(5) = 0

Understanding zeros is a starting point for both graphing polynomial functions and solving polynomial equations. Each zero represents an x-intercept or a place where the graph crosses or touches the x-axis. Recognizing the pattern of zeros helps in predicting the shape and orientation of the graph of the polynomial.

Factors of Polynomials

A factor of a polynomial is a polynomial that divides another polynomial without leaving a remainder. In the context of zeros, each zero of a polynomial corresponds to a factor.

Given the zeros

• -4
• 1
• 5

, these can translate into factors by rearranging them into expressions of the form

• (x + 4)
• (x - 1)
• (x - 5)

Then, you can express the polynomial as a product of these factors. This gives us a factored form of the polynomial, which is crucial for understanding how polynomials are constructed from their roots. In essence, if you multiply these factors, you reconstruct the polynomial. Recognizing and writing polynomials in their factored form is an effective method for solving polynomial equations and simplifies the process of finding zeros.

Expanding Polynomials

Expanding a polynomial involves multiplying out its factors to express it in standard form. The standard form is when the polynomial is written as a sum of terms ordered by their decreasing powers of the variable.

Taking the factored polynomial function \[(x + 4)(x - 1)(x - 5)\], the process of expansion begins:

• First, multiply the first two factors:\[(x + 4)(x - 1) = x^2 + 3x - 4\]
• Then, take this result and multiply it by the third factor: \[(x^2 + 3x - 4)(x - 5)\]
• Distribute each term, which can be simplified as:\[x^3 - 2x^2 - 19x + 20\]

This final expression is the expanded form of the polynomial. Expanding polynomials is key to applying certain operations like addition, subtraction, and derivation. It also allows for better interpretation and analysis, especially when the polynomial is involved in further mathematical evaluations or applications.