(5x4−2x3−7x2−39)÷(x2+2x−4)=

Jasonbradley

5 min read

·

Jun 16, 2024

27

Share

Polynomial Long Division: (5x⁴ - 2x³ - 7x² - 39) ÷ (x² + 2x - 4)

This article will walk you through the process of dividing the polynomial (5x⁴ - 2x³ - 7x² - 39) by (x² + 2x - 4) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in elementary school, but instead of dividing numbers, we are dividing polynomials. The goal is to find the quotient and remainder of the division.

Steps to Perform Polynomial Long Division

1. Set up the division. Write the dividend (5x⁴ - 2x³ - 7x² - 39) inside the division symbol and the divisor (x² + 2x - 4) outside.

        ____________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
   

2. Divide the leading terms. Focus on the leading terms of the dividend and the divisor (5x⁴ and x²). Ask yourself: "What do I need to multiply x² by to get 5x⁴?" The answer is 5x². Write this term above the division symbol, aligned with the x⁴ term.

        5x² _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
   

3. Multiply the divisor. Multiply the divisor (x² + 2x - 4) by the term you just wrote (5x²). This gives you 5x⁴ + 10x³ - 20x². Write this result below the dividend, aligning terms with corresponding powers.

        5x² _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
                5x⁴ + 10x³ - 20x²
   

4. Subtract. Subtract the expression you just wrote from the dividend. Remember to change the signs of the terms you are subtracting.

        5x² _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
                5x⁴ + 10x³ - 20x²
                -----------------
                    -12x³ + 13x² - 39 
   

5. Bring down the next term. Bring down the next term of the dividend (-39).

        5x² _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
                5x⁴ + 10x³ - 20x²
                -----------------
                    -12x³ + 13x² - 39 
   

6. Repeat steps 2-5. Now focus on the new leading terms: -12x³ and x². What do you need to multiply x² by to get -12x³? The answer is -12x. Write -12x above the division symbol, aligned with the x³ term.

        5x² - 12x _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
                5x⁴ + 10x³ - 20x²
                -----------------
                    -12x³ + 13x² - 39 
   

7. Continue until the degree of the remainder is less than the degree of the divisor. Repeat steps 2-5 until the degree of the polynomial you get after subtraction is less than the degree of the divisor (x² + 2x - 4).

        5x² - 12x + 19 _________
   x² + 2x - 4 | 5x⁴ - 2x³ - 7x² - 39
                5x⁴ + 10x³ - 20x²
                -----------------
                    -12x³ + 13x² - 39 
                    -12x³ - 24x² + 48x
                    --------------------
                         37x² + 48x - 39
                         37x² + 74x - 148
                         --------------------
                               -26x + 109
   

Result

The result of the division is:

Quotient: 5x² - 12x + 19

Remainder: -26x + 109

Therefore, we can write:

(5x⁴ - 2x³ - 7x² - 39) ÷ (x² + 2x - 4) = 5x² - 12x + 19 + (-26x + 109) / (x² + 2x - 4)