%2F(x-2).jpeg "(5x+6x^3-8)/(x-2)")
Polynomial Division: (5x + 6x³ - 8) / (x - 2)
This article will guide you through the process of dividing the polynomial (5x + 6x³ - 8) by the binomial (x - 2) using polynomial long division.
Step 1: Set Up the Division
First, arrange the dividend (5x + 6x³ - 8) and the divisor (x - 2) in a long division format. Remember to include any missing terms with a coefficient of 0.
_________
x - 2 | 6x³ + 0x² + 5x - 8
Step 2: Divide the Leading Terms
Divide the leading term of the dividend (6x³) by the leading term of the divisor (x). This gives us 6x². Write this result above the dividend.
6x²
x - 2 | 6x³ + 0x² + 5x - 8
Step 3: Multiply and Subtract
Multiply the result (6x²) by the divisor (x - 2) and write the product below the dividend.
6x²
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
Subtract the product from the dividend.
6x²
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
---------
12x² + 5x
Step 4: Repeat the Process
Bring down the next term of the dividend (5x). Now, repeat steps 2 and 3 with the new polynomial (12x² + 5x).
Divide the leading term (12x²) by the leading term of the divisor (x). This gives us 12x.
6x² + 12x
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
---------
12x² + 5x
12x² - 24x
Subtract the product from the new polynomial.
6x² + 12x
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
---------
12x² + 5x
12x² - 24x
---------
29x - 8
Step 5: Repeat Again
Bring down the last term of the dividend (-8) and repeat steps 2 and 3.
Divide the leading term (29x) by the leading term of the divisor (x). This gives us 29.
6x² + 12x + 29
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
---------
12x² + 5x
12x² - 24x
---------
29x - 8
29x - 58
Subtract the product from the new polynomial.
6x² + 12x + 29
x - 2 | 6x³ + 0x² + 5x - 8
6x³ - 12x²
---------
12x² + 5x
12x² - 24x
---------
29x - 8
29x - 58
---------
50
Step 6: The Result
Since the degree of the remainder (50) is less than the degree of the divisor (x - 2), we stop here.
Therefore, the result of the division is:
6x² + 12x + 29 + 50/(x - 2)
This means: (5x + 6x³ - 8) / (x - 2) = 6x² + 12x + 29 + 50/(x - 2)