Polynomial long division is a method used to divide one polynomial by another, similar to how you divide numbers. Below are the step-by-step instructions for performing polynomial long division:

### Step 1: Arrange in Standard Form
Make sure both the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by) are written in descending order of their degrees. If any terms are missing, add them with a zero coefficient.

### Step 2: Divide the Leading Terms
Look at the leading term (the term with the highest degree) of the dividend and the leading term of the divisor. Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.

### Step 3: Multiply and Subtract
Multiply the entire divisor by the first term of the quotient you just found. Subtract this product from the dividend to get a new polynomial.

### Step 4: Repeat
Take the new polynomial from step 3 as your new dividend. Repeat steps 2 and 3, dividing the new leading term by the leading term of the divisor, multiplying the divisor by this new term, and subtracting again.

### Step 5: Continue Until Remainder
Keep repeating the process until the degree of the remainder (the polynomial left after subtracting) is less than the degree of the divisor. At this point, the quotient you've found is the answer, and the remaining polynomial is the remainder.

### Example:
Divide \( 2x^3 + 3x^2 - 5x + 6 \) by \( x - 1 \).

1. **Divide:** \( \frac{2x^3}{x} = 2x^2 \).
2. **Multiply:** \( 2x^2(x - 1) = 2x^3 - 2x^2 \).
3. **Subtract:** \( (2x^3 + 3x^2 - 5x + 6) - (2x^3 - 2x^2) = 5x^2 - 5x + 6 \).
4. **Repeat:** \( \frac{5x^2}{x} = 5x \).
5. **Multiply:** \( 5x(x - 1) = 5x^2 - 5x \).
6. **Subtract:** \( (5x^2 - 5x + 6) - (5x^2 - 5x) = 6 \).
7. **Final Step:** \( \frac{6}{x - 1} \) results in a remainder of 6, since \( x - 1 \) doesn't divide evenly into 6.

So, the result is \( 2x^2 + 5x + 1 \) with a remainder of 6.

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### Do you want more details or have any specific questions?

Here are some related questions:
1. How do you handle polynomial division when the divisor is a higher degree than the dividend?
2. What happens when there's a remainder in polynomial division?
3. Can polynomial long division be applied to non-linear divisors?
4. How does synthetic division differ from polynomial long division?
5. How does polynomial division relate to factorization?

#### Tip: Always organize terms by degree and ensure that every degree is accounted for, even if the coefficient is zero.

Learn the step-by-step process for polynomial long division with a detailed example. This guide covers key algebraic concepts such as dividing polynomials, handling remainders, and applying the polynomial division algorithm. Suitable for students in Grades 9-12.

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