Questions: Find the quotient and remainder using long division for [ frac2 x^3-8 x^2+7 x-172 x^2+5 ] The quotient is The remainder is

Solution Steps

To solve the problem of finding the quotient and remainder using long division for the given polynomials, follow these steps:

1. Set up the long division by dividing the leading term of the numerator by the leading term of the denominator.
2. Multiply the entire divisor by this result and subtract from the original polynomial.
3. Repeat the process with the new polynomial formed after subtraction until the degree of the remainder is less than the degree of the divisor.
4. The quotient will be the sum of all the results obtained in each step, and the remainder will be the final polynomial left after the last subtraction.

We start with the polynomial division of the numerator \( 2x^3 - 8x^2 + 7x - 17 \) by the denominator \( 2x^2 + 5 \).

Using polynomial long division, we divide the leading term of the numerator \( 2x^3 \) by the leading term of the denominator \( 2x^2 \), which gives us \( x \).

Next, we multiply the entire denominator \( 2x^2 + 5 \) by \( x \): \[ x(2x^2 + 5) = 2x^3 + 5x \]

We then subtract this result from the original numerator: \[ (2x^3 - 8x^2 + 7x - 17) - (2x^3 + 5x) = -8x^2 + 2x - 17 \]

Now, we repeat the process with the new polynomial \( -8x^2 + 2x - 17 \). We divide the leading term \( -8x^2 \) by \( 2x^2 \), which gives us \( -4 \).

Next, we multiply the entire denominator \( 2x^2 + 5 \) by \( -4 \): \[ -4(2x^2 + 5) = -8x^2 - 20 \]

We subtract this from the current polynomial: \[ (-8x^2 + 2x - 17) - (-8x^2 - 20) = 2x + 3 \]

At this point, the degree of the remainder \( 2x + 3 \) is less than the degree of the denominator \( 2x^2 + 5 \). Therefore, we stop here.

The quotient is \( x - 4 \) and the remainder is \( 2x + 3 \).

Final Answer