Use synthetic division to divide p(x) = 3x3 − 10x2 − 31x − 12 by (3x + 5). Then check the solution.
A handy synthetic division calculator can be found here:
http://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
You will have to adjust your divisor to account for the leading 3.
To use synthetic division, follow these steps:
1. Write down the coefficients of the polynomial in descending order: p(x) = 3x^3 - 10x^2 - 31x - 12. The coefficient of each term corresponds to the power of x.
2. Determine the divisor: (3x + 5). To use synthetic division, we need to convert this divisor into a binomial of the form (x - a), where 'a' is the opposite sign of the constant term in the divisor. So, (3x + 5) becomes (x - (-5/3)) or (x + 5/3).
3. Set up the synthetic division table. Place the coefficients of the polynomial (3, -10, -31, -12) in the top row of the table, followed by the constant term of the divisor (5/3) in the side column.
| 3 -10 -31 -12
-----------------------
5/3|
4. Bring down the first coefficient, which is 3, below the line.
| 3 -10 -31 -12
-----------------------
5/3| 3
5. Multiply the divisor by the first value you brought down and write the result below the second coefficient.
| 3 -10 -31 -12
-----------------------
5/3| 3 -5
6. Add the second coefficient with the result. Write the sum below the third coefficient.
| 3 -10 -31 -12
-----------------------
5/3| 3 -5 -26
7. Multiply the divisor by the value you just wrote down and write the result below the fourth coefficient.
| 3 -10 -31 -12
-----------------------
5/3| 3 -5 -26
__________________
-10 -40 -12
8. Add the third coefficient with the result and write the sum below the last coefficient.
| 3 -10 -31 -12
-----------------------
5/3| 3 -5 -26
__________________
-10 -40 -12
______
-52
The final row represents the coefficients of the quotient polynomial: 3x^2 - 5x - 26. The result is interpreted as p(x) = (3x + 5)(3x^2 - 5x - 26).
To check the solution, you can multiply the divisor (3x + 5) by the quotient polynomial (3x^2 - 5x - 26) using the distributive property:
(3x + 5)(3x^2 - 5x - 26) = 9x^3 - 15x^2 - 78x + 15x^2 - 25x - 130 - 78x + 130
Combining like terms:
9x^3 - 78x - 25x - 78x - 5x + 15x^2 + 130 - 130
Simplifying:
9x^3 - 106x + 15x^2
This result matches the original polynomial p(x) = 3x^3 - 10x^2 - 31x - 12, confirming that the synthetic division was performed correctly.