Use synthetic division to divide the polynomial 2x3 – 45x + 28 by x + 5, and write the quotient polynomial and the remainder. [Be careful – notice that there is no x2 term.]. Show work.
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To perform synthetic division, we need to set up the synthetic division table.
Let's divide the polynomial 2x^3 - 45x + 28 by x + 5.
-5 | 2 0 -45 28
Step 1: Bring down the coefficient of the highest degree term.
2
Step 2: Multiply the divisor (x + 5) by the result in Step 1.
-5 * 2 = -10
Step 3: Add the product from Step 2 to the coefficient of the next term.
0 + -10 = -10
Step 4: Multiply the divisor (x + 5) by the result in Step 3.
-5 * -10 = 50
Step 5: Add the product from Step 4 to the coefficient of the next term.
-45 + 50 = 5
Step 6: Multiply the divisor (x + 5) by the result in Step 5.
-5 * 5 = -25
Step 7: Add the product from Step 6 to the coefficient of the next term.
28 + -25 = 3
Now we have completed the synthetic division.
The quotient polynomial is 2x^2 - 10x + 5, and the remainder is 3.
To use synthetic division, we follow these steps:
Step 1: Arrange the polynomial in descending order, with missing terms filled in with zero coefficients. In this case, we have:
2x^3 - 45x + 28
Step 2: Determine the divisor. In this case, the divisor is x + 5.
Step 3: Flip the sign of the divisor and write it as -5.
Step 4: Set up the synthetic division table by writing the coefficients of the polynomial as the first row, and then writing the -5 (divisor) in the leftmost column. We will ignore the x term since there is no x^2 term.
-5 | 2 0 -45 28
Step 5: Bring down the first coefficient (2) directly below the horizontal line.
-5 | 2
|
|
|
|
Step 6: Multiply the divisor (-5) by the number below the line (2), and write the result under the next coefficient (-45).
-5 | 2
| -10
|
|
|
Step 7: Add the two numbers in the second row (0 + (-10) = -10), and place the result under the next coefficient (-45).
-5 | 2
| -10
| -10
|
|
Step 8: Repeat Steps 6 and 7 until all coefficients have been processed.
-5 | 2
| -10
| -10
| 10
|
Step 9: The number below the line in the last row (10) is the remainder.
Step 10: The numbers in the second row (2, -10, -10) are the coefficients of the quotient polynomial.
Therefore, the quotient polynomial is 2x^2 - 10x - 10, and the remainder is 10.