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(v^3-2v^2-14v-5)/(v+3)

5 min read Jun 16, 2024
(v^3-2v^2-14v-5)/(v+3)

Dividing Polynomials: (v^3-2v^2-14v-5) / (v+3)

This article will guide you through the process of dividing the polynomial (v^3-2v^2-14v-5) by (v+3). We'll use polynomial long division to achieve this.

Understanding Polynomial Long Division

Polynomial long division is very similar to the long division you learned in elementary school. The main goal is to find the quotient and remainder when dividing one polynomial by another.

Step-by-Step Division

  1. Set up the division: Write the dividend (v^3-2v^2-14v-5) inside the division symbol and the divisor (v+3) outside.

         _________
v+3 | v^3 - 2v^2 - 14v - 5

  2. Divide the leading terms: Focus on the leading terms of the dividend (v^3) and the divisor (v). Ask yourself: "What do I multiply 'v' by to get 'v^3'?" The answer is v^2. Write v^2 above the division symbol, aligning it with the v^2 term in the dividend.

         v^2      
v+3 | v^3 - 2v^2 - 14v - 5

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

         v^2      
v+3 | v^3 - 2v^2 - 14v - 5 
     -(v^3 + 3v^2)
     ----------------
           -5v^2 - 14v

  4. Bring down the next term: Bring down the next term of the dividend (-14v).

         v^2      
v+3 | v^3 - 2v^2 - 14v - 5 
     -(v^3 + 3v^2)
     ----------------
           -5v^2 - 14v - 5

  5. Repeat steps 2-4: Repeat the process, focusing on the new leading term (-5v^2) and the divisor's leading term (v). Ask: "What do I multiply 'v' by to get '-5v^2'?" The answer is -5v. Write -5v above the division symbol, aligning it with the v term in the dividend.

         v^2 - 5v     
v+3 | v^3 - 2v^2 - 14v - 5 
     -(v^3 + 3v^2)
     ----------------
           -5v^2 - 14v - 5
           -(-5v^2 - 15v)
           ---------------
                    v - 5

  6. Continue the process: Repeat steps 2-4 again. This time, we focus on the new leading term (v) and the divisor's leading term (v). Ask: "What do I multiply 'v' by to get 'v'?" The answer is 1. Write 1 above the division symbol, aligning it with the constant term in the dividend.

         v^2 - 5v + 1     
v+3 | v^3 - 2v^2 - 14v - 5 
     -(v^3 + 3v^2)
     ----------------
           -5v^2 - 14v - 5
           -(-5v^2 - 15v)
           ---------------
                    v - 5
                    -(v + 3)
                    ---------
                           -8

  7. Identify the quotient and remainder: We have reached the end of the division. The quotient is the polynomial above the division symbol: v^2 - 5v + 1. The remainder is the last term: -8.

Conclusion

Therefore, the division of (v^3-2v^2-14v-5) by (v+3) can be expressed as:

(v^3-2v^2-14v-5) / (v+3) = v^2 - 5v + 1 - 8/(v+3)

This means that (v^3-2v^2-14v-5) is equal to (v+3) multiplied by (v^2 - 5v + 1) plus the remainder (-8).

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