Polynomial Long Division: Find the Quotient and Remainder of (x⁴-5x³+x-4) / (x²-7x+1)

Polynomial Long Division: Dividing (x⁴ - 5x³ + x - 4) by (x² - 7x + 1)

This tutorial explains how to perform polynomial long division to find the quotient and remainder when dividing (x⁴ - 5x³ + x - 4) by (x² - 7x + 1).

Here's a step-by-step breakdown:

1. Set up the division: Write the dividend (x⁴ - 5x³ + x - 4) and the divisor (x² - 7x + 1) as you would in a regular long division problem. Remember to include placeholders (0x³ and 0x²) for any missing terms in the dividend.

x² - 7x + 1 ___________________x⁴ - 5x³ + 0x² + x - 4 | x⁴ + 0x³ + 0x² + x - 4

2. Divide leading terms: Divide the leading term of the dividend (x⁴) by the leading term of the divisor (x²). The result, x², is the first term of the quotient.

x² - 7x + 1 ___________________x⁴ - 5x³ + 0x² + x - 4 | x⁴ + 0x³ + 0x² + x - 4 - (x⁴ - 7x³ + x²)

3. Multiply and subtract: Multiply the divisor (x² - 7x + 1) by the first term of the quotient (x²) and subtract the result from the dividend.

x² - 7x + 1 ___________________x⁴ - 5x³ + 0x² + x - 4 | x⁴ + 0x³ + 0x² + x - 4 - (x⁴ - 7x³ + x²) ___________________ 2x³ - x² + x - 4

4. Repeat steps 2 and 3: Bring down the next term of the dividend (-x²) and repeat steps 2 and 3 with the new polynomial (2x³ - x² + x - 4).

x² - 7x + 1 ___________________x⁴ - 5x³ + 0x² + x - 4 | x⁴ + 0x³ + 0x² + x - 4 - (x⁴ - 7x³ + x²) ___________________ 2x³ - x² + x - 4 - (2x³ - 14x² + 2x) ___________________ 13x² - x - 4

5. Continue the process: Repeat steps 2 and 3 with the new polynomial (13x² - x - 4).

x² - 7x + 1 ___________________x⁴ - 5x³ + 0x² + x - 4 | x⁴ + 0x³ + 0x² + x - 4 - (x⁴ - 7x³ + x²) ___________________ 2x³ - x² + x - 4 - (2x³ - 14x² + 2x) ___________________ 13x² - x - 4 - (13x² - 91x + 13) ___________________ 90x - 176. Identify the quotient and remainder: The degree of the new polynomial (90x - 17) is less than the degree of the divisor, so we stop here. * The quotient is the polynomial written above the division line: x² + 2x + 13. * The remainder is the final polynomial at the bottom: 90x - 17.

Therefore, the result of dividing (x⁴ - 5x³ + x - 4) by (x² - 7x + 1) is:

Quotient: x² + 2x + 13 Remainder: 90x - 17