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.
