(2x^3+5x^2+9) Divided By (x+3)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x^3 + 5x^2 + 9) ÷ (x + 3)

This article will guide you through the process of dividing the polynomial (2x^3 + 5x^2 + 9) by (x + 3) using polynomial long division.

Polynomial Long Division

Polynomial long division is similar to long division with numbers. Here's how it works:

1. Set up the division: Write the dividend (2x^3 + 5x^2 + 9) inside the division symbol and the divisor (x + 3) outside.

       ____________
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
   

2. Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives 2x^2. Write this above the division symbol.

       2x^2        
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
   

3. Multiply and subtract: Multiply the divisor (x + 3) by the term you just wrote (2x^2). This gives 2x^3 + 6x^2. Write this result below the dividend and subtract.

       2x^2        
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          -(2x^3 + 6x^2)
          ----------------
                 -x^2 + 0x 
   

4. Bring down the next term: Bring down the next term of the dividend (0x).

       2x^2        
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          -(2x^3 + 6x^2)
          ----------------
                 -x^2 + 0x 
   

5. Repeat steps 2-4: Repeat the process of dividing, multiplying, and subtracting. Divide the new leading term (-x^2) by the leading term of the divisor (x), which gives -x. Write this above the division symbol. Multiply (x + 3) by -x and subtract.

       2x^2 - x       
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          -(2x^3 + 6x^2)
          ----------------
                 -x^2 + 0x 
                 -(-x^2 - 3x)
                 ---------------
                          3x + 9
   

6. Continue until the degree of the remainder is less than the degree of the divisor: Bring down the next term (9) and repeat the process.

       2x^2 - x + 3    
   x + 3 | 2x^3 + 5x^2 + 0x + 9 
          -(2x^3 + 6x^2)
          ----------------
                 -x^2 + 0x 
                 -(-x^2 - 3x)
                 ---------------
                          3x + 9 
                          -(3x + 9)
                          ----------
                                 0
   

Result

The remainder is 0. Therefore, (2x^3 + 5x^2 + 9) divided by (x + 3) is 2x^2 - x + 3.

We can express this result as:

(2x^3 + 5x^2 + 9) ÷ (x + 3) = 2x^2 - x + 3