Problem 80 - Algebra A Combined Function

Question

Write an equation that has three solutions, \(0,1,\) and 2

Step-by-Step Solution

Answer

The equation is \(x^3 - 3x^2 + 2x = 0\).

Step 1: Understanding the Problem

We need to find a polynomial equation that has three solutions: 0, 1, and 2. This means the equation will have roots at these points.

Step 2: Using the Factor Theorem

According to the factor theorem, if \(a\) is a root of a polynomial equation, then \(x-a\) is a factor of the polynomial. Thus, the factors for our roots would be \(x-0\), \(x-1\), and \(x-2\).

Step 3: Writing the Equation

Combine the factors from each root into a single polynomial equation. Start by writing the expression as \((x-0)(x-1)(x-2) = 0\).

Step 4: Simplifying the Equation

Now expand the expression to find the polynomial. Simplify \((x)(x-1)(x-2)\). First, multiply \((x-1)(x-2)\): \[(x-1)(x-2) = x^2 - 3x + 2\] Then, multiply the result by \(x\): \[x(x^2 - 3x + 2) = x^3 - 3x^2 + 2x\]

Step 5: Final Equation

The polynomial equation that has solutions 0, 1, and 2 is then given by:\[x^3 - 3x^2 + 2x = 0\]

Key Concepts

Factor Theorem

The Factor Theorem is an essential tool in algebra for dealing with polynomial equations. It provides a connection between factors and roots of polynomials. In simple terms, it says that if a polynomial has a root at a certain point, there is a corresponding linear factor. Let's break this down:

• If a polynomial \(f(x)\) has a root \(a\), then it can be expressed as: \(f(x) = (x - a) \cdot Q(x)\), where \(Q(x)\) is another polynomial.
• This means that substituting \(a\) back into the polynomial makes it zero, i.e., \(f(a) = 0\).

In our example, we are given roots 0, 1, and 2. So, the polynomial must include the factors \(x-0\), \(x-1\), and \(x-2\). Expanding these will help form our polynomial equation. Factorizing a polynomial is essentially working backward using the Factor Theorem.

Roots of a Polynomial

The roots of a polynomial are the values of \(x\) for which the polynomial equals zero. These values are also known as solutions or zeros of the polynomial. Finding the roots is a fundamental aspect of solving polynomial equations. Here's a simple guideline:

• If you have a polynomial \( P(x) \, \, \ =\ \, a_nx^n + ... + a_1x + a_0\), finding its roots involves solving the equation \( P(x) = 0\).
• The number of roots is determined by the degree of the polynomial. For instance, a cubic polynomial (degree 3) can have up to 3 real roots.

In the exercise we discussed, the roots given are 0, 1, and 2. These roots directly indicate where the polynomial will intersect the x-axis when graphed. Knowing the roots helps in constructing the polynomial.

Multiplying Polynomials

When constructing a polynomial from its factors, multiplying polynomials comes into play. It is the process that combines individual factors into a single polynomial expression. Here's how you can do it:

• Start by multiplying pairs of factors to form intermediate expressions. For example, multiply \(x-1\) and \(x-2\): \[ (x - 1)(x - 2) = x^2 - 3x + 2 \]
• Then, continue multiplying the resulting expression by any remaining factors. In the current exercise, multiply the resulting quadratic expression by \(x\): \[ x(x^2 - 3x + 2) = x^3 - 3x^2 + 2x \]

The result is a single polynomial which encompasses all the roots as demonstrated by its factorization. Multiplying polynomials helps us understand the structure of the polynomial and check for its correctness relative to the given roots.