(4x^3-7x^2-11x+5)/(4x+5) Long Division

Jasonbradley

5 min read

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Jun 16, 2024

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Long Division of Polynomials: (4x³ - 7x² - 11x + 5) / (4x + 5)

Long division of polynomials is a process used to divide a polynomial by another polynomial of a lower degree. This process is similar to the long division of numbers.

Here's how to divide (4x³ - 7x² - 11x + 5) by (4x + 5):

Step 1: Set up the division.

Write the dividend (4x³ - 7x² - 11x + 5) inside the division symbol and the divisor (4x + 5) outside the symbol.

        ____________
4x + 5 | 4x³ - 7x² - 11x + 5 

Step 2: Divide the leading terms.

Divide the leading term of the dividend (4x³) by the leading term of the divisor (4x). This gives us x². Write x² above the dividend.

        x² _________
4x + 5 | 4x³ - 7x² - 11x + 5 

Step 3: Multiply the quotient by the divisor.

Multiply the quotient (x²) by the divisor (4x + 5). This gives us 4x³ + 5x². Write this below the dividend.

        x² _________
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²

Step 4: Subtract.

Subtract the product from the dividend. This gives us -12x². Bring down the next term (-11x).

        x² _________
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 

Step 5: Repeat the process.

Repeat steps 2-4 with the new dividend (-12x² - 11x).

• Divide the leading term of the new dividend (-12x²) by the leading term of the divisor (4x). This gives us -3x. Write -3x above the dividend.

        x² - 3x ______
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 

• Multiply the new quotient (-3x) by the divisor (4x + 5). This gives us -12x² - 15x. Write this below the new dividend.

        x² - 3x ______
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 
                -12x² - 15x

• Subtract the product from the new dividend. This gives us 4x. Bring down the next term (+5).

        x² - 3x ______
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 
                -12x² - 15x
                ---------
                        4x + 5

Step 6: Repeat again.

Repeat steps 2-4 with the new dividend (4x + 5).

• Divide the leading term of the new dividend (4x) by the leading term of the divisor (4x). This gives us 1. Write 1 above the dividend.

        x² - 3x + 1 __
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 
                -12x² - 15x
                ---------
                        4x + 5

• Multiply the new quotient (1) by the divisor (4x + 5). This gives us 4x + 5. Write this below the new dividend.

        x² - 3x + 1 __
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 
                -12x² - 15x
                ---------
                        4x + 5
                        4x + 5

• Subtract the product from the new dividend. This gives us 0.

        x² - 3x + 1 __
4x + 5 | 4x³ - 7x² - 11x + 5 
           4x³ + 5x²
           ---------
                -12x² - 11x 
                -12x² - 15x
                ---------
                        4x + 5
                        4x + 5
                        ---------
                                0

Therefore, the result of dividing (4x³ - 7x² - 11x + 5) by (4x + 5) is x² - 3x + 1.

This can also be written as:

(4x³ - 7x² - 11x + 5) / (4x + 5) = x² - 3x + 1