91Ó°ÊÓ

Long division is used to divide polynomials when the following conditions are met: 1. The divisor is a polynomial with a degree greater than or equal to 1. 2. The degree of the divisor is less than the degree of the dividend. 3. The division is not a simple algebraic simplification or a factorization.

Let's consider a division of two polynomials: Dividend: \(P(x) = 3x^3 - 5x^2 + 2x + 4\) Divisor: \(D(x) = x - 1\) Since the degree of the divisor (1) is less than the degree of the dividend (3), we can use long division to divide these polynomials.

To perform long division between polynomials, follow these steps: 1. Write the dividend and divisor in a long division format. 2. Divide the first term of the dividend by the first term of the divisor. Write the result as the first term of the quotient. 3. Multiply the divisor by the first term of the quotient, write the result below the dividend, and draw a line under it. 4. Subtract the result from step 3 from the dividend. 5. Drop the next term from the original dividend. 6. Repeat steps 2-5 until all terms of the dividend have been used. Let's perform this for our example problem. Step 1: ``` _______________ x - 1 | 3x^3 - 5x^2 + 2x + 4 ``` Step 2: Divide the first term of the dividend (3x³) by the first term of the divisor (x): \( \frac{3x^3}{x} = 3x^2 \) Write 3x² as the first term of the quotient. Step 3: Multiply the divisor by the first term of the quotient: \( (x - 1)(3x^2) = 3x^3 - 3x^2 \) Step 4: Subtract the result from the dividend. ``` 3x^2 _________ x - 1 | 3x^3 - 5x^2 + 2x + 4 -(3x^3 - 3x^2) __________________ -2x^2 ``` Step 5: Drop the next term (+ 2x) from the dividend. ``` 3x^2 _________ x - 1 | 3x^3 - 5x^2 + 2x + 4 -(3x^3 - 3x^2) __________________ -2x^2 + 2x ``` Now we repeat steps 2-5: Step 2: Divide -2x² by x: \( \frac{-2x^2}{x} = -2x \) Write -2x as the next term of the quotient. Step 3: Multiply the divisor by the second term of the quotient: \( (x - 1)(-2x) = -2x^2 + 2x \) Step 4: Subtract the result from the remaining dividend. ``` 3x^2 - 2x ________ x - 1 | 3x^3 - 5x^2 + 2x + 4 -(3x^3 - 3x^2) __________________ -2x^2 + 2x -(-2x^2 + 2x) __________________ 4 ``` There are no more terms left to drop, so the long division is complete, and we have: Quotient: \(Q(x) = 3x^2 - 2x\) Remainder: \(R(x) = 4\) So, the result of the division is: \( \frac{3x^3 - 5x^2 + 2x + 4}{x - 1} = 3x^2 - 2x + \frac{4}{x-1} \) In conclusion, we use long division to divide polynomials when the divisor has a degree between 1 and the degree of the dividend, and the division is not a simple algebraic simplification or factorization.