(a^2-28) Divided By (a-5)

3 min read Jun 16, 2024

Dividing (a^2 - 28) by (a - 5)

This problem involves dividing a polynomial by a binomial. We can use long division to solve it. Here's how:

Step 1: Set up the Long Division

          a + 5    
a - 5 | a^2 + 0a - 28

We write the dividend (a^2 - 28) inside the division symbol. Notice that we've added a 0a term as a placeholder for the missing linear term.
The divisor (a - 5) is written outside the division symbol.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend (a^2) by the leading term of the divisor (a). This gives us a.
Write a above the division symbol, aligned with the a^2 term.

          a + 5    
a - 5 | a^2 + 0a - 28
          a^2 - 5a

Step 3: Multiply and Subtract

Multiply the divisor (a - 5) by the term we just wrote above (a). This gives us a^2 - 5a.
Write this result below the dividend.
Subtract the two polynomials:

          a + 5    
a - 5 | a^2 + 0a - 28
          a^2 - 5a
          -------
                  5a - 28

Step 4: Bring Down the Next Term

Bring down the next term from the dividend (-28).

          a + 5    
a - 5 | a^2 + 0a - 28
          a^2 - 5a
          -------
                  5a - 28

Step 5: Repeat Steps 2-4

Divide the leading term of the new polynomial (5a) by the leading term of the divisor (a). This gives us 5.
Write 5 above the division symbol, aligned with the constant term.
Multiply the divisor (a - 5) by 5. This gives us 5a - 25.
Subtract the two polynomials:

          a + 5    
a - 5 | a^2 + 0a - 28
          a^2 - 5a
          -------
                  5a - 28
                  5a - 25
                  -------
                         -3

Step 6: Remainder

We're left with a remainder of -3.

The Result

Therefore, (a^2 - 28) divided by (a - 5) is a + 5 with a remainder of -3. We can express this as:

(a^2 - 28) / (a - 5) = a + 5 - 3/(a - 5)