(3x^3-17x^2+15x-25)/(x-5) Long Division

Jasonbradley

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·

Jun 16, 2024

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Long Division of Polynomials: (3x^3 - 17x^2 + 15x - 25) / (x - 5)

Long division of polynomials is a process used to divide one polynomial by another. It is similar to the long division of numbers, but with polynomials. Let's break down the process of dividing (3x^3 - 17x^2 + 15x - 25) by (x - 5).

Steps:

1. Set up the division: Write the dividend (3x^3 - 17x^2 + 15x - 25) inside the division symbol and the divisor (x - 5) outside.

        ___________
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
   

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

        3x^2 _______
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
   

3. Multiply the divisor: Multiply the divisor (x - 5) by the result we just obtained (3x^2). This gives us 3x^3 - 15x^2. Write this product below the dividend.

        3x^2 _______
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
   

4. Subtract: Subtract the product from the dividend. This gives us -2x^2 + 15x. Bring down the next term of the dividend (-25).

        3x^2 _______
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
           ----------
                -2x^2 + 15x - 25 
   

5. Repeat steps 2-4: Repeat the process, dividing the new leading term (-2x^2) by the leading term of the divisor (x). This gives us -2x. Write this result above the division symbol.

        3x^2 - 2x ______
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
           ----------
                -2x^2 + 15x - 25 
                -2x^2 + 10x
   

6. Multiply and subtract: Multiply the divisor (x - 5) by -2x and subtract the product from the previous result. This gives us 5x - 25.

        3x^2 - 2x ______
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
           ----------
                -2x^2 + 15x - 25 
                -2x^2 + 10x
                ----------
                       5x - 25
   

7. Repeat steps 2-4: Again, divide the new leading term (5x) by the leading term of the divisor (x). This gives us 5. Write this result above the division symbol.

        3x^2 - 2x + 5 ____
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
           ----------
                -2x^2 + 15x - 25 
                -2x^2 + 10x
                ----------
                       5x - 25
                       5x - 25
   

8. Subtract: Subtract the product from the previous result. We get a remainder of 0.

        3x^2 - 2x + 5 ____
   x - 5 | 3x^3 - 17x^2 + 15x - 25 
           3x^3 - 15x^2
           ----------
                -2x^2 + 15x - 25 
                -2x^2 + 10x
                ----------
                       5x - 25
                       5x - 25
                       ------
                            0
   

Result:

Therefore, (3x^3 - 17x^2 + 15x - 25) / (x - 5) = 3x^2 - 2x + 5

Since the remainder is 0, we say that (x - 5) is a factor of (3x^3 - 17x^2 + 15x - 25).