Question

divide the polynomial (2x^4 + 3x^3 + 3x^2 - 5x - 3) by (2x^2 - x - 1)

Explanation:

Step1: Divide the leading terms

Divide the leading term of the numerator \(2x^4\) by the leading term of the denominator \(2x^2\) to get \(x^2\).

Step2: Multiply the denominator by \(x^2\)

Multiply \(2x^2 - x - 1\) by \(x^2\) to get \(2x^4 - x^3 - x^2\).

Step3: Subtract from the numerator

Subtract \(2x^4 - x^3 - x^2\) from \(2x^4 + 3x^3 + 3x^2 - 5x - 3\):

$$
LATEXBLOCK0
$$

Step4: Divide the new leading term

Divide the leading term of \(4x^3\) by \(2x^2\) to get \(2x\).

Step5: Multiply the denominator by \(2x\)

Multiply \(2x^2 - x - 1\) by \(2x\) to get \(4x^3 - 2x^2 - 2x\).

Step6: Subtract from the new polynomial

Subtract \(4x^3 - 2x^2 - 2x\) from \(4x^3 + 4x^2 - 5x - 3\):

$$
LATEXBLOCK1
$$

Step7: Divide the new leading term

Divide the leading term of \(6x^2\) by \(2x^2\) to get \(3\).

Step8: Multiply the denominator by \(3\)

Multiply \(2x^2 - x - 1\) by \(3\) to get \(6x^2 - 3x - 3\).

Step9: Subtract to get the remainder

Subtract \(6x^2 - 3x - 3\) from \(6x^2 - 3x - 3\) to get a remainder of \(0\).

Answer:

\(x^2 + 2x + 3\)