Questions: Divide x^3 + x^2 - 5x - 2 by x - 2

Solution Steps

To divide the polynomial \(x^3 + x^2 - 5x - 2\) by \(x - 2\), we can use polynomial long division or synthetic division. Here, we'll use synthetic division because the divisor is in the form \(x - c\). The steps are as follows:

1. Identify the coefficients of the dividend polynomial: \(1, 1, -5, -2\).
2. Use the root of the divisor, which is \(2\), for synthetic division.
3. Perform synthetic division to find the quotient and remainder.

We are given the polynomial \( P(x) = x^3 + x^2 - 5x - 2 \) and we need to divide it by the linear polynomial \( D(x) = x - 2 \).

Using synthetic division with the root of the divisor \( x - 2 \), which is \( 2 \), we set up the coefficients of the polynomial:

• Coefficients: \( [1, 1, -5, -2] \)

We perform the synthetic division:

1. Bring down the leading coefficient \( 1 \).
2. Multiply \( 1 \) by \( 2 \) and add to the next coefficient \( 1 \) to get \( 3 \).
3. Multiply \( 3 \) by \( 2 \) and add to the next coefficient \( -5 \) to get \( 1 \).
4. Multiply \( 1 \) by \( 2 \) and add to the last coefficient \( -2 \) to get \( 0 \).

The result of the synthetic division gives us:

• Quotient: \( Q(x) = x^2 + 3x + 1 \)
• Remainder: \( R = 0 \)

The division can be expressed as: \[ P(x) = (x - 2)(x^2 + 3x + 1) + 0 \] This indicates that \( P(x) \) is exactly divisible by \( D(x) \).

Final Answer