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.