Question

Using long division determine which of the fallowing polynomials divides evenly for a divisor of x + 3

Answer 1

To determine if a polynomial divides evenly by x + 3 using long division, we divide the polynomial by x + 3 and check if the remainder is zero.

Let's take an example.

Suppose we have the polynomial 2x^3 + 5x^2 - 4x - 8 and we want to check if it divides evenly by x + 3.

-2x^2 + 11x + 29
_________________________
x + 3 | 2x^3 + 5x^2 - 4x - 8

1. We start by dividing the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives us 2x^2.
2. We multiply the entire divisor (x + 3) by the result from step 1 (2x^2) and write it under the dividend.
This gives us 2x^2(x + 3) = 2x^3 + 6x^2.
3. We subtract this product from the dividend: (2x^3 + 5x^2 - 4x - 8) - (2x^3 + 6x^2). Simplifying, we get -x^2 - 4x - 8.
4. We bring down the next term, which is -4x.
5. We repeat steps 1-4 with the new dividend (-x^2 - 4x - 8).
- Divide (-x^2) by x to get -x.
- Multiply (x + 3) by -x to get -x(x + 3) = -x^2 - 3x.
- Subtract this product from the new dividend: (-x^2 - 4x - 8) - (-x^2 - 3x). Simplifying, we get -x - 8.
6. Bring down the next term, which is -8.
7. Repeat steps 1-4 with the new dividend (-x - 8).
- Divide (-x) by x to get -1.
- Multiply (x + 3) by -1 to get -(x + 3) = -x - 3.
- Subtract this product from the new dividend: (-x - 8) - (-(x + 3)). Simplifying, we get -5.
8. We have reached the end of the polynomial, so we can say that 2x^3 + 5x^2 - 4x - 8 divided by x + 3 leaves a remainder of -5.

Since the remainder is not zero, the polynomial 2x^3 + 5x^2 - 4x - 8 does not divide evenly by x + 3.