## AI Polynomial Division Solver

### Polynomial Division

#### Introduction to Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps simplify expressions and solve polynomial equations.

#### Types of Polynomial Division

1. Long Division

2. Synthetic Division (for specific cases where the divisor is a binomial of the form \(x – c\))

#### Long Division of Polynomials

**Steps for Long Division:**

1. Arrange Polynomials:

– Write both the dividend and the divisor in descending order of their degrees.

– For example, divide \(2x^3 + 3x^2 – 5x + 6\) by \(x – 2\).

2. Divide the Leading Terms:

– Divide the first term of the dividend by the first term of the divisor.

– \( \frac{2x^3}{x} = 2x^2 \)

3. Multiply and Subtract:

– Multiply the entire divisor by the result from step 2 and subtract this from the original dividend.

– \( (2x^3 + 3x^2 – 5x + 6) – (2x^2(x – 2)) = (2x^3 + 3x^2 – 5x + 6) – (2x^3 – 4x^2) = 7x^2 – 5x + 6 \)

4. Repeat:

– Repeat steps 2 and 3 with the new polynomial \(7x^2 – 5x + 6\).

– \( \frac{7x^2}{x} = 7x \)

– Multiply and subtract: \( (7x^2 – 5x + 6) – (7x(x – 2)) = (7x^2 – 5x + 6) – (7x^2 – 14x) = 9x + 6 \)

5. Continue Until Completion:

– Repeat the process until the degree of the remainder is less than the degree of the divisor.

– \( \frac{9x}{x} = 9 \)

– Multiply and subtract: \( (9x + 6) – (9(x – 2)) = (9x + 6) – (9x – 18) = 24 \)

6. Combine Results:

– The quotient is \( 2x^2 + 7x + 9 \) and the remainder is \( 24 \).

– The final result is \( 2x^2 + 7x + 9 + \frac{24}{x – 2} \).

#### Synthetic Division

When to Use:

– Synthetic division is used when the divisor is a binomial in the form \(x – c\).

#### Steps for Synthetic Division:

1. Set Up:

– Write down the coefficients of the dividend polynomial.

– For \(2x^3 + 3x^2 – 5x + 6\) and divisor \(x – 2\), write \(2, 3, -5, 6\).

2. Use the Zero of the Divisor:

– The zero of \(x – 2\) is \(2\).

– Write \(2\) to the left and a vertical line to separate.

3. Bring Down the Leading Coefficient:

– Bring down the first coefficient directly below the line.

– \(2\)

4. Multiply and Add:

– Multiply the leading coefficient by the zero of the divisor and write the result below the next coefficient, then add.

– \(2 \times 2 = 4\)

– \(3 + 4 = 7\)

– Continue this process:

– \(7 \times 2 = 14\)

– \(-5 + 14 = 9\)

– \(9 \times 2 = 18\)

– \(6 + 18 = 24\)

5. Interpret the Result:

– The result below the line represents the coefficients of the quotient polynomial.

– For the example: \(2x^2 + 7x + 9\) and the remainder \(24\).

6. Combine Results:

– The final result is \(2x^2 + 7x + 9 + \frac{24}{x – 2}\).

#### Practice Problems

1. Divide \(4x^3 + 6x^2 – x + 5\) by \(2x – 1\) using long division.

2. Divide \(3x^4 – 2x^3 + x – 4\) by \(x + 1\) using synthetic division.

3. Divide \(5x^3 + 3x^2 – x – 2\) by \(x – 3\) using long division.