Polynomial Division: (x^3 + 6x + 8) ÷ (x - 2)
Hey guys! Let's dive into dividing polynomials, specifically focusing on how to divide the polynomial (x^3 + 6x + 8) by (x - 2). We'll explore two common methods: polynomial long division and synthetic division. Both are super useful, but synthetic division is often quicker when dividing by a linear factor like (x - 2). So, grab your pencils, and let's get started!
Polynomial Long Division
Polynomial long division is similar to the long division you learned back in elementary school, but instead of numbers, we're working with polynomials. It's a step-by-step process that helps us break down the division. Let's walk through it with our example, (x^3 + 6x + 8) ÷ (x - 2).
Step 1: Set Up the Division
First, set up the long division. Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial we're dividing by) outside.
x - 2 | x^3 + 0x^2 + 6x + 8
Notice that we included a 0x^2 term in the dividend. This is super important! We need to include placeholders for any missing terms to keep everything aligned correctly. In this case, since there's no x^2 term in the original polynomial, we add 0x^2.
Step 2: Divide the Leading Terms
Next, divide the leading term of the dividend (x^3) by the leading term of the divisor (x).
x^3 / x = x^2
Write x^2 above the division symbol, aligned with the x^2 term.
x^2
x - 2 | x^3 + 0x^2 + 6x + 8
Step 3: Multiply and Subtract
Multiply the divisor (x - 2) by the term we just wrote above (x^2).
x^2 * (x - 2) = x^3 - 2x^2
Write this result below the dividend and subtract.
x^2
x - 2 | x^3 + 0x^2 + 6x + 8
-(x^3 - 2x^2)
x^2
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
Step 4: Bring Down the Next Term
Bring down the next term from the dividend (+6x).
x^2
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
Step 5: Repeat the Process
Now, repeat the process. Divide the leading term of the new dividend (2x^2) by the leading term of the divisor (x).
2x^2 / x = 2x
Write +2x above the division symbol, aligned with the x term.
x^2 + 2x
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
Multiply the divisor (x - 2) by 2x.
2x * (x - 2) = 2x^2 - 4x
Write this result below and subtract.
x^2 + 2x
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-(2x^2 - 4x)
x^2 + 2x
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-2x^2 + 4x
-------------
10x + 8
Step 6: Bring Down the Last Term and Repeat
Bring down the last term from the dividend (+8).
x^2 + 2x
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-2x^2 + 4x
-------------
10x + 8
Divide the leading term of the new dividend (10x) by the leading term of the divisor (x).
10x / x = 10
Write +10 above the division symbol, aligned with the constant term.
x^2 + 2x + 10
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-2x^2 + 4x
-------------
10x + 8
Multiply the divisor (x - 2) by 10.
10 * (x - 2) = 10x - 20
Write this result below and subtract.
x^2 + 2x + 10
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-2x^2 + 4x
-------------
10x + 8
-(10x - 20)
x^2 + 2x + 10
x - 2 | x^3 + 0x^2 + 6x + 8
- x^3 + 2x^2
-------------
2x^2 + 6x
-2x^2 + 4x
-------------
10x + 8
-10x + 20
-------------
28
Step 7: Determine the Quotient and Remainder
The quotient is the polynomial above the division symbol (x^2 + 2x + 10), and the remainder is the value left over after the final subtraction (28).
So, (x^3 + 6x + 8) ÷ (x - 2) = x^2 + 2x + 10 + 28/(x - 2).
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - a). It's generally quicker and more efficient than long division, especially when dealing with simple divisors. Let's use synthetic division to divide (x^3 + 6x + 8) by (x - 2).
Step 1: Set Up the Synthetic Division
Write down the coefficients of the dividend. Remember to include a 0 for any missing terms. In our case, the coefficients are 1 (for x^3), 0 (for x^2), 6 (for x), and 8 (the constant term).
Write the value of a from the divisor (x - a). In our case, x - 2, so a = 2.
Set up the synthetic division like this:
2 | 1 0 6 8
|_________
Step 2: Bring Down the First Coefficient
Bring down the first coefficient (which is 1) below the line.
2 | 1 0 6 8
|_________
1
Step 3: Multiply and Add
Multiply the value of a (which is 2) by the number you just brought down (which is 1).
2 * 1 = 2
Write the result (2) under the next coefficient (which is 0).
2 | 1 0 6 8
| 2
|_________
1
Add the numbers in the second column (0 + 2 = 2). Write the sum (2) below the line.
2 | 1 0 6 8
| 2
|_________
1 2
Step 4: Repeat the Process
Repeat the process. Multiply the value of a (which is 2) by the number you just wrote below the line (which is 2).
2 * 2 = 4
Write the result (4) under the next coefficient (which is 6).
2 | 1 0 6 8
| 2 4
|_________
1 2
Add the numbers in the third column (6 + 4 = 10). Write the sum (10) below the line.
2 | 1 0 6 8
| 2 4
|_________
1 2 10
Step 5: Repeat One Last Time
Repeat the process one last time. Multiply the value of a (which is 2) by the number you just wrote below the line (which is 10).
2 * 10 = 20
Write the result (20) under the last coefficient (which is 8).
2 | 1 0 6 8
| 2 4 20
|_________
1 2 10
Add the numbers in the last column (8 + 20 = 28). Write the sum (28) below the line.
2 | 1 0 6 8
| 2 4 20
|_________
1 2 10 28
Step 6: Determine the Quotient and Remainder
The numbers below the line, except for the last one, are the coefficients of the quotient. The last number is the remainder.
In our case, the coefficients are 1, 2, and 10. This means the quotient is x^2 + 2x + 10. The remainder is 28.
So, (x^3 + 6x + 8) ÷ (x - 2) = x^2 + 2x + 10 + 28/(x - 2).
Conclusion
Whether you choose polynomial long division or synthetic division, you'll arrive at the same answer. Synthetic division is often quicker for linear divisors, but long division is more versatile for more complex divisors. Both are essential tools in your polynomial-manipulating arsenal. Keep practicing, and you'll become a pro in no time! I hope this breakdown helps you to understand these concepts better. Happy dividing, folks! Remember that including the placeholder terms are also very important. Good luck!