Create an easy Polynomial equation and solve for the sum, the difference, and the product

Answer 1

Let's start with a simple polynomial equation. We can consider the polynomial \( P(x) = x^2 + 3x + 2 \).

To find the roots of the equation, we set it to zero:

\[ x^2 + 3x + 2 = 0 \]

We can factor this polynomial:

\[ (x + 1)(x + 2) = 0 \]

Setting each factor to zero gives us the roots:

1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

Now that we have the roots \( x_1 = -1 \) and \( x_2 = -2 \), we can find the sum, the difference, and the product of these roots.

1. Sum of the roots:

\[ \text{Sum} = x_1 + x_2 = -1 + (-2) = -3 \]

2. Difference of the roots:

\[ \text{Difference} = x_1 - x_2 = -1 - (-2) = -1 + 2 = 1 \]

3. Product of the roots:

\[ \text{Product} = x_1 \times x_2 = (-1) \times (-2) = 2 \]

Summary:

• Sum of the roots: \(-3\)
• Difference of the roots: \(1\)
• Product of the roots: \(2\)

Answer 2

Explain

Answer 3

Sure! Let's break down the polynomial equation and the steps we took to find the sum, difference, and product of its roots.

Step-by-Step Explanation

1. Choosing a Polynomial: We started with a simple polynomial equation, \( P(x) = x^2 + 3x + 2 \). A polynomial is an expression consisting of variables raised to various powers, and here we have a quadratic polynomial (degree 2).

2. Setting the Polynomial to Zero: To find the roots of the polynomial (the values of \( x \) that make \( P(x) = 0 \)), we set the equation equal to zero: \[ x^2 + 3x + 2 = 0 \]

3. Factoring the Polynomial: We factored the quadratic expression. This means finding two binomials that multiply together to produce the original polynomial. In this case, we identified: \[ (x + 1)(x + 2) = 0 \] We can verify this because when we Multiply: \[ (x + 1)(x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2 \]

4. Finding the Roots: We then set each factor equal to zero:

   • For \( x + 1 = 0 \) → \( x = -1 \)
   • For \( x + 2 = 0 \) → \( x = -2 \)

   Thus, the roots of the polynomial are \( x_1 = -1 \) and \( x_2 = -2 \).

5. Calculating the Sum: The sum of the roots (\( x_1 + x_2 \)) is calculated as follows: \[ -1 + (-2) = -1 - 2 = -3 \]

6. Calculating the Difference: The difference of the roots (\( x_1 - x_2 \)) is calculated as follows: \[ -1 - (-2) = -1 + 2 = 1 \] Here, we are subtracting the second root from the first, and the negative double negative turns into addition.

7. Calculating the Product: The product of the roots (\( x_1 \times x_2 \)) is calculated as follows: \[ -1 \times -2 = 2 \] The product of two negative numbers is positive.

Summary of Results

• The sum of the roots is \(-3\).
• The difference of the roots is \(1\).
• The product of the roots is \(2\).

Importance of Roots

Roots of polynomial equations are important in various fields of mathematics and applied sciences, including algebra, calculus, and beyond. Knowing how to find and calculate the relationships of roots can help in solving more complex problems.

Feel free to ask if you have any more questions!