Problem 72 - Algebra 2

Question

Construct a polynomial function with the given zeros. $$ x=-2,-1,3 $$

Step-by-Step Solution

Answer

The polynomial function with the given zeros \(x = -2, -1, 3\) is \(f(x) = (x + 2)(x + 1)(x - 3)\).

Step 1: Identify the Zeros

The polynomial has zeros at \(x = -2, -1, 3\). These zeros will be used to construct the factors of the polynomial function.

Step 2: Construct Factors from Zeros

The standard factor corresponding to a zero \(x_i\) is \((x - x_i)\). So the factors corresponding to the zeros at \(x = -2, -1, 3\) will be \((x + 2)\), \((x + 1)\), and \((x - 3)\), respectively.

Step 3: Combine Factors into Polynomial

A polynomial's standard form is a sum of terms, each of which consists of a coefficient and one or more variables raised to a nonnegative integer exponent. The factors can be combined by multiplication to form a polynomial: \(f(x) = (x + 2)(x + 1)(x - 3)\).

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are crucial because they are the points where the polynomial evaluates to zero. In other words, if we plug a zero into the polynomial, the result is zero. These zeros are also known as roots or solutions of the polynomial equation. For example, if we have a polynomial function \(f(x)\) with zeros at \(x = -2, -1,\) and \(3\), it means:

• \(f(-2) = 0\)
• \(f(-1) = 0\)
• \(f(3) = 0\)

Zeros help us to understand the behavior of the polynomial on the graph, as these are the points where the line or curve intersects the x-axis. Knowing the zeros allows us to construct the polynomial by working with its factors. This is why identifying zeros is often the first step when dealing with polynomial construction and solving.

Factorization

Factorization involves breaking down a polynomial into simpler components called factors, which when multiplied together give back the original polynomial. For polynomials, factors are derived from algebraic expressions that correspond to the zeros of the polynomial. Here is how it works:Suppose we have zeros \(x = -2, -1,\) and \(3\). The factors become:

• \(x + 2\) for the zero \(x = -2\)
• \(x + 1\) for the zero \(x = -1\)
• \(x - 3\) for the zero \(x = 3\)

Factorization is crucial because it allows us to express the polynomial in a form that makes it easy to solve and analyze. Multiplying these factors together gives us the polynomial \(f(x) = (x + 2)(x + 1)(x - 3)\). This factorized form is easy to understand and manipulate, making it handy for solving equations or graphing the polynomial function.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. In the realm of polynomials, these expressions take on a special character, allowing us to represent and solve complex equations. Each factor of a polynomial is in itself an algebraic expression.When constructing a polynomial from its zeros, we deal with simple expressions like \((x - a)\), where \(a\) is a zero. For the polynomial with zeros \(-2, -1,\) and \(3\):

• \((x + 2)\), \((x + 1)\), and \((x - 3)\) are the basic algebraic expressions formed
• Each expression corresponds to subtracting a zero of the polynomial from \(x\).

These basic components are then multiplied to form the polynomial \(f(x) = (x + 2)(x + 1)(x - 3)\), demonstrating how algebraic expressions provide a structured way to build and understand polynomial functions. This organized structure is an essential part of algebra that facilitates the solving of equations, comprehension of functions, and the visualization of graphs.