%2F(x-6).jpeg "(x^4-5x^3-8x^2+13x-12)/(x-6)")
Solving the Polynomial Division: (x^4 - 5x^3 - 8x^2 + 13x - 12) / (x - 6)
This article aims to demonstrate the process of dividing the polynomial x^4 - 5x^3 - 8x^2 + 13x - 12 by the binomial x - 6. We will utilize the long division method to find the quotient and remainder.
Step 1: Setting Up the Long Division
Begin by setting up the problem as a long division problem:
____________
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
Step 2: Dividing the Leading Terms
- The leading term of the divisor (x - 6) is x.
- The leading term of the dividend (x^4 - 5x^3 - 8x^2 + 13x - 12) is x^4.
- Divide x^4 by x, which gives us x^3. Write this above the line in the quotient space.
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
Step 3: Multiplying the Quotient Term
- Multiply the quotient term (x^3) by the divisor (x - 6). This gives us x^4 - 6x^3.
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
Step 4: Subtracting the Result
- Subtract the result (x^4 - 6x^3) from the dividend. Remember to change the signs of the terms being subtracted.
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3
Step 5: Bringing Down the Next Term
- Bring down the next term of the dividend (-8x^2).
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
Step 6: Repeating the Process
- Repeat steps 2-5 with the new dividend (x^3 - 8x^2).
- Divide the leading term of the new dividend (x^3) by the leading term of the divisor (x), which gives us x^2.
- Write this term above the line in the quotient space.
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
- Multiply the new quotient term (x^2) by the divisor (x - 6), resulting in x^3 - 6x^2.
- Subtract this product from the current dividend, remembering to change the signs.
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2
- Bring down the next term of the dividend (+13x).
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
Step 7: Continuing the Division
- Repeat the process with the new dividend (-2x^2 + 13x).
- Divide the leading term of the dividend (-2x^2) by the leading term of the divisor (x), which gives us -2x.
- Write this term above the line in the quotient space.
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
- Multiply the new quotient term (-2x) by the divisor (x - 6), resulting in -2x^2 + 12x.
- Subtract this product from the current dividend, remembering to change the signs.
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x
Step 8: Final Steps
- Bring down the last term of the dividend (-12).
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
- Divide the leading term of the new dividend (x) by the leading term of the divisor (x), which gives us 1.
- Write this term above the line in the quotient space.
x^3 + x^2 - 2x + 1
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
x - 6
- Multiply the new quotient term (1) by the divisor (x - 6), resulting in x - 6.
- Subtract this product from the current dividend, remembering to change the signs.
x^3 + x^2 - 2x + 1
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
x - 6
--------
-6
Step 9: The Result
The remainder of the division is -6. Therefore, the final result of the polynomial division can be expressed as:
(x^4 - 5x^3 - 8x^2 + 13x - 12) / (x - 6) = x^3 + x^2 - 2x + 1 - 6/(x - 6)
This can also be written as:
(x^4 - 5x^3 - 8x^2 + 13x - 12) = (x - 6)(x^3 + x^2 - 2x + 1) - 6