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

5 min read Jun 16, 2024

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

This article will demonstrate how to perform polynomial long division on the expression (4x^3-7x^2-11x+5)/(4x+5).

Step 1: Setting up the Division

First, set up the division problem. The dividend (4x^3-7x^2-11x+5) goes inside the division symbol, and the divisor (4x+5) goes outside.

             _______
4x+5 | 4x^3-7x^2-11x+5

Step 2: Dividing the Leading Terms

We begin by dividing the leading term of the dividend (4x^3) by the leading term of the divisor (4x). This gives us x^2. Write x^2 above the division symbol, aligning it with the x^3 term.

             x^2      
4x+5 | 4x^3-7x^2-11x+5

Step 3: Multiplying and Subtracting

Next, multiply the divisor (4x+5) by the term we just wrote (x^2). This gives us 4x^3 + 5x^2. Write this result below the dividend, aligning terms with the same powers of x.

             x^2      
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)

Now, subtract the entire expression we just wrote from the dividend. This leaves us with -12x^2 -11x.

             x^2      
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x

Step 4: Repeating the Process

Repeat the process from step 2. Divide the leading term of the new dividend (-12x^2) by the leading term of the divisor (4x), which gives us -3x. Write -3x above the division symbol, aligning it with the x^2 term.

             x^2 -3x    
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x

Multiply the divisor (4x+5) by -3x, which gives us -12x^2 -15x. Write this result below the previous line, aligning terms with the same powers of x.

             x^2 -3x    
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x 
           -(-12x^2-15x)

Now, subtract the entire expression we just wrote from the previous line. This leaves us with 4x+5.

             x^2 -3x    
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x 
           -(-12x^2-15x)
           -------------
                 4x+5

Step 5: Final Step

Repeat the process one more time. Divide the leading term of the new dividend (4x) by the leading term of the divisor (4x), which gives us 1. Write 1 above the division symbol, aligning it with the constant term.

             x^2 -3x +1  
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x 
           -(-12x^2-15x)
           -------------
                 4x+5 
                 -(4x+5)

Multiply the divisor (4x+5) by 1, which gives us 4x+5. Write this result below the previous line.

Subtract the entire expression we just wrote from the previous line. This leaves us with 0.

             x^2 -3x +1  
4x+5 | 4x^3-7x^2-11x+5 
       -(4x^3+5x^2)
       -------------
           -12x^2-11x 
           -(-12x^2-15x)
           -------------
                 4x+5 
                 -(4x+5) 
                 -------
                     0

Solution

We have reached a remainder of 0. Therefore, the solution to the division problem is:

(4x^3-7x^2-11x+5)/(4x+5) = x^2 - 3x + 1