Answer :

Answer:Certainly! If you're looking to solve a polynomial equation, I'll guide you through the general steps. Polynomials are equations involving variables raised to non-negative integer powers. Here's how we approach solving a polynomial equation step-by-step:

### Step-by-Step Solution for Polynomial Equation:

1. **Understand the Equation:**

  - Identify the polynomial equation. It usually takes the form \( P(x) = 0 \), where \( P(x) \) is the polynomial expression.

2. **Determine the Degree:**

  - Find the highest power of \( x \) in the polynomial. This gives you the degree of the polynomial.

3. **Factorization (if applicable):**

  - If the polynomial can be factored, factor it completely. This step may involve techniques like factoring by grouping, using the difference of squares, or recognizing special patterns.

4. **Use of Varies

Step-by-step explanation:Certainly! Let's go through a step-by-step explanation of how to solve a polynomial equation. I'll provide a general outline that you can follow for most polynomial equations.

### Step-by-Step Explanation for Solving a Polynomial Equation:

Let's say we have a polynomial equation \( P(x) = 0 \).

1. **Understand the Polynomial Equation:**

  - Identify the polynomial equation given. It will typically be in the form \( P(x) = 0 \), where \( P(x) \) is the polynomial expression.

2. **Determine the Degree of the Polynomial:**

  - Find the highest power of \( x \) in the polynomial equation. This highest power is the degree of the polynomial.

3. **Attempt to Factor the Polynomial (if possible):**

  - If the polynomial can be factored, factor it completely. Factoring involves expressing the polynomial as a product of simpler polynomials.

  - Techniques for factoring include:

    - **Common Factor:** Look for a common factor among all terms.

    - **Grouping:** Group terms together and factor out common factors from each group.

    - **Special Forms:** Recognize special forms like difference of squares (\( a^2 - b^2 \)) or perfect squares (\( a^2 + 2ab + b^2 \)).

4. **Use the Rational Root Theorem (if necessary):**

  - If the polynomial is not easily factorable, you can use the Rational Root Theorem to find possible rational roots. The theorem states that any rational root \( \frac{p}{q} \) of the polynomial \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 \) must have \( p \) as a factor of the constant term \( a_0 \) and \( q \) as a factor of the leading coefficient \( a_n \).

5. **Apply the Remainder Theorem (if applicable):**

  - The Remainder Theorem states that if you divide a polynomial \( P(x) \) by \( x - c \), the remainder is \( P(c) \). This theorem can help you check potential roots or simplify calculations.

6. **Solve for \( x \):**

  - Once you have potential roots (zeros) from factoring or from the Rational Root Theorem, substitute them back into the original polynomial equation \( P(x) = 0 \) to solve for \( x \).

7. **Verify Solutions:**

  - After finding potential solutions, verify each one by substituting it back into the original polynomial equation to ensure it satisfies \( P(x) = 0 \).

8. **Complex Solutions (if any):**

  - For higher degree polynomials, solutions might involve complex numbers. Express complex solutions in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit.

### Example:

Let's work through a simple example to illustrate these steps:

**Example:** Solve the equation \( x^2 - 5x + 6 = 0 \).

**Solution:**

1. **Identify the Polynomial Equation:**

  - \( x^2 - 5x + 6 = 0 \)

2. **Determine the Degree:**

  - The highest power of \( x \) is 2, so the degree of the polynomial is 2.

3. **Factor the Polynomial:**

  - The polynomial \( x^2 - 5x + 6 \) can be factored as \( (x - 2)(x - 3) = 0 \).

4. **Find Solutions:**

  - Set each factor equal to zero:

    - \( x - 2 = 0 \) gives \( x = 2 \)

    - \( x - 3 = 0 \) gives \( x = 3 \)

5. **Verify Solutions:**

  - Substitute \( x = 2 \) and \( x = 3 \) back into the original equation \( x^2 - 5x + 6 = 0 \) to verify both satisfy the equation.

Therefore, the solutions to the equation \( x^2 - 5x + 6 = 0 \) are \( x = 2 \) and \( x = 3 \).

This example demonstrates the step-by-step process of solving a quadratic polynomial equation. Adjust the approach according to the degree of the polynomial and the specific form of the equation you are working with.