Divide \frac{\left(10x^4+20x^3-31x^2+x+19\ right)}{\left(x+3\right)} (x+3)
(10x
4
+20x
3
−31x
2
+x+19)

. What is the Remainder?

R = 7

Explanation:

You can divide a polynomial by “x - b” using synthetic division. (See photo below; work is on the left, form is on the right).
Set up a chart with “-b” in the top left corner.
In the top row, list the coefficients in the order that appears with the polynomial.
Write “–“ and “X” to remind you to subtract and multiply.

1. Bring down the first dividend coefficient (or 10)
2. Multiply the first answer coefficient by -b (or 3)
3. Write the product (or 30) under the second coefficient
4. Subtract in the second column to find the second answer coefficient (-10)
5. Repeat until the last subtraction, which is the remainder (R = 7)

*** You could also solve using the remainder theorem by substituting x = -3 into the dividend and solving.