Questions: divide x^4+3 x^2-2 x-1 by (x+1) using long division

Solution Steps

To divide the polynomial \(x^{4} + 3x^{2} - 2x - 1\) by \(x + 1\) using long division, we follow these steps:

1. Divide the leading term of the dividend by the leading term of the divisor.
2. Multiply the entire divisor by the result from step 1 and subtract from the dividend.
3. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

We are dividing the polynomial \( P(x) = x^{4} + 3x^{2} - 2x - 1 \) by \( D(x) = x + 1 \).

Divide the leading term of \( P(x) \) by the leading term of \( D(x) \): \[ \frac{x^{4}}{x} = x^{3} \] Multiply \( D(x) \) by \( x^{3} \): \[ x^{3}(x + 1) = x^{4} + x^{3} \] Subtract from \( P(x) \): \[ (x^{4} + 3x^{2} - 2x - 1) - (x^{4} + x^{3}) = -x^{3} + 3x^{2} - 2x - 1 \]

Now divide the leading term of the new polynomial by the leading term of \( D(x) \): \[ \frac{-x^{3}}{x} = -x^{2} \] Multiply \( D(x) \) by \( -x^{2} \): \[ -x^{2}(x + 1) = -x^{3} - x^{2} \] Subtract: \[ (-x^{3} + 3x^{2} - 2x - 1) - (-x^{3} - x^{2}) = 4x^{2} - 2x - 1 \]

Divide the leading term again: \[ \frac{4x^{2}}{x} = 4x \] Multiply \( D(x) \) by \( 4x \): \[ 4x(x + 1) = 4x^{2} + 4x \] Subtract: \[ (4x^{2} - 2x - 1) - (4x^{2} + 4x) = -6x - 1 \]

Divide the leading term: \[ \frac{-6x}{x} = -6 \] Multiply \( D(x) \) by \( -6 \): \[ -6(x + 1) = -6x - 6 \] Subtract: \[ (-6x - 1) - (-6x - 6) = 5 \]

The quotient is \( Q(x) = x^{3} - x^{2} + 4x - 6 \) and the remainder is \( R = 5 \). Thus, we can express the division as: \[ \frac{P(x)}{D(x)} = Q(x) + \frac{R}{D(x)} = x^{3} - x^{2} + 4x - 6 + \frac{5}{x + 1} \]

Final Answer