Problem 19 - Algebra Form and Function

Question

Find possible formulas for the polynomials described. The degree is \(n=2\) and the zeros are \(x=2,-3\).

Step-by-Step Solution

Answer

Question: Find a possible formula for a polynomial of degree 2 with zeros at x=2 and x=-3. Answer: P(x) = x^2 + x - 6

Step 1: Determine the factors from the zeros

Based on the zeros, we can create two factors corresponding to each root. We write the factors in the form \((x-a)\), where \(a\) is the root. So, the factors are \((x-2)\) and \((x+3)\).

Step 2: Multiply the factors

Now, we multiply the factors to get the polynomial formula. In this case, we multiply \((x-2)(x+3)\): \((x-2)(x+3) = x^2 + 3x - 2x - 6\)

Step 3: Simplify the expression

After multiplying the factors, we can simplify the polynomial by combining like terms: \(x^2 + 3x - 2x - 6 = x^2 + x - 6\)

Step 4: Write the final polynomial

The possible polynomial formula for the given information is: \(P(x) = x^2 + x - 6\)

Key Concepts

Degree of a Polynomial

Polynomials are mathematical expressions that consist of variables and coefficients. One of the core concepts in understanding polynomials is the **degree of a polynomial**.

The degree of a polynomial can be defined as the highest power of the variable in the expression. It indicates the number of roots, or solutions, a polynomial can have.

In the given exercise, the polynomial has a degree of 2, denoted as \(n=2\). This suggests that the polynomial is quadratic, characterized by a term with the square of a variable, \(x^2\). Quadratic polynomials usually have two roots or zeros, as shown in this problem.

To identify the degree of a polynomial, you simply look for the term with the largest exponent in the polynomial expression. This provides critical information about the behavior and graph of the polynomial.

Polynomial Zeros

The **zeros of a polynomial** are the values of \(x\) that make the polynomial equal to zero.

Finding these zeros is essential because they provide insight into the polynomial's characteristics. In the problem, the zeros are given as \(x=2\) and \(x=-3\). Each zero corresponds to a point where the graph of the polynomial crosses, or touches, the \(x\)-axis.

Understanding how to work with zeros involves knowing that they are solutions to the equation \(P(x) = 0\).

- They help in constructing polynomials by forming factors.
- Each zero creates a linear factor in the form \((x-a)\), where \(a\) is the zero.

This feature of polynomials allows us to build them step-by-step from their zero values, which in practical terms means that knowing the zeros directly helps in composing the polynomial.

Factoring Polynomials

**Factoring polynomials** involves breaking down a polynomial into simpler terms or factors that can be multiplied together to yield the original polynomial.

In this exercise, given zeros \(x=2\) and \(x=-3\), these correspond to the factors \((x-2)\) and \((x+3)\), respectively. Factoring is crucial for simplifying polynomials and solving equations.

The process:

• Identify the zeros of the polynomial.
• Create factors that correspond to these zeros.
• Multiply the factors to get the polynomial.

By factoring, you not only simplify expressions but also solve polynomial equations and analyze their graphs. Using the example, multiplying the factors \((x-2)(x+3)\), you uncover the polynomial formula \(x^2 + x - 6\). This shows how factoring is a reverse process of expanding a polynomial.

Simplifying Expressions

**Simplifying expressions** is about reducing polynomials to their simplest form for ease of use and clarity.

After multiplying factors to expand them into polynomials, like in our exercise, it's necessary to combine like terms.

- Look for terms with the same power of the variable.
- Add or subtract coefficients of these like terms.

For the expression \(x^2 + 3x - 2x - 6\), simplifying means combining the \(3x\) and \(-2x\) terms, which results in \(x^2 + x - 6\). This simplified form is crucial because it provides the cleanest representation of the polynomial, making subsequent mathematical operations like integration, differentiation, or graphing straightforward.