Question

use long division, determine which of the following polynomials divides evenly for a divisor of x+3

Answer 1

To determine which polynomial divides evenly by \(x + 3\) using long division, we generally need to perform polynomial long division. However, you haven't provided any specific polynomials to test. Let's go through the process with a generic polynomial \(P(x)\) and a divisor \(D(x) = x + 3\).

Here is how the process of long division works:

1. **Setup the Division**: Write \(P(x)\) (the polynomial you want to divide) under the long division symbol and \(x + 3\) (the divisor) outside.

2. **Divide the Leading Terms**: Divide the leading term of \(P(x)\) by the leading term of \(D(x)\). This result will be the first term of the quotient.

3. **Multiply**: Multiply \(D(x)\) by the term you just found in the quotient.

4. **Subtract**: Subtract this product from \(P(x)\) to get a new polynomial.

5. **Repeat**: Repeat the process with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor \(x + 3\).

6. **Determine Divisibility**: If the remainder is zero, then \(P(x)\) divides evenly by \(x + 3\).

To demonstrate, let's say we want to check if a polynomial such as \(P(x) = x^2 + 3x + 4\) is divisible by \(x + 3\).

1. **Setup**:
\[
\text{Divide } x^2 + 3x + 4 \text{ by } x + 3
\]

2. **Divide the Leading Terms**:
\[
x^2 \div x = x
\]

3. **Multiply**:
\[
x(x + 3) = x^2 + 3x
\]

4. **Subtract**:
\[
(x^2 + 3x + 4) - (x^2 + 3x) = 4
\]

Now we have a remainder of \(4\). Since the remainder is not zero, \(x^2 + 3x + 4\) does not divide evenly by \(x + 3\).

If you provide specific polynomials, I can perform the long division for each and check for divisibility!