Questions: Use long division to divide the polynomials. (2v^4 + 3v^3 - 10v^2 - 3v + 6) / (v^2 - 2)

Solution Steps

To divide the polynomials using long division, we will follow these steps:

1. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
2. Multiply the entire divisor by this term and subtract the result 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 tasked with dividing the polynomial \( 2v^4 + 3v^3 - 10v^2 - 3v + 6 \) by \( v^2 - 2 \).

We divide the leading term of the dividend \( 2v^4 \) by the leading term of the divisor \( v^2 \): \[ \frac{2v^4}{v^2} = 2v^2 \] This gives us the first term of the quotient.

Next, we multiply the entire divisor \( v^2 - 2 \) by \( 2v^2 \): \[ 2v^2(v^2 - 2) = 2v^4 - 4v^2 \] Now, we subtract this from the original polynomial: \[ (2v^4 + 3v^3 - 10v^2 - 3v + 6) - (2v^4 - 4v^2) = 3v^3 - 6v^2 - 3v + 6 \]

We repeat the process with the new polynomial \( 3v^3 - 6v^2 - 3v + 6 \). Divide the leading term \( 3v^3 \) by \( v^2 \): \[ \frac{3v^3}{v^2} = 3v \] This gives us the second term of the quotient.

Multiply the divisor by \( 3v \): \[ 3v(v^2 - 2) = 3v^3 - 6v \] Subtract this from the current polynomial: \[ (3v^3 - 6v^2 - 3v + 6) - (3v^3 - 6v) = -6v^2 + 3v + 6 \]

Now, we divide \( -6v^2 \) by \( v^2 \): \[ \frac{-6v^2}{v^2} = -6 \] This gives us the third term of the quotient.

Multiply the divisor by \(-6\): \[ -6(v^2 - 2) = -6v^2 + 12 \] Subtract this from the current polynomial: \[ (-6v^2 + 3v + 6) - (-6v^2 + 12) = 3v - 6 \] Now, the degree of the remainder \( 3v - 6 \) is less than the degree of the divisor \( v^2 - 2 \).

Final Answer