Problem 47 - College Algebra

Question

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

Step-by-Step Solution

Answer

The result of the long division of \(2x^{3} - 3x^{2} - 11x + 7\) by \(x-3\) is \(2x^{2} + 3x + 2 + \frac{1}{x-3}\).

Step 1: Divide

First, divide the leading term in the dividend, \(2x^{3}\), by the leading term in the divisor, \(x\). This gives you \(2x^{2}\), the first term of the quotient.

Step 2: Multiply

Next, multiply the divisor \(x-3\) by the first term of the quotient \(2x^{2}\). This gives \(2x^{3} - 6x^{2}\). Write this under the dividend aligned by like terms.

Step 3: Subtract

Subtract the result from the previous step (\(2x^{3} - 6x^{2}\)) from the first two terms of the dividend. This involves subtracting each term individually. Then, we get \((-3x^{2}) - (-6x^{2}) = 3x^{2}\).

Step 4: Bring Down

To continue the division, bring down the next term of the dividend, which is \(-11x\). Now the new dividend for the next round of division is \(3x^{2} - 11x\). Repeat steps 1-4 until you've brought down all terms.

Step 5: Continue Division

You will continue with the divisions until the degree of the remainder is less than the degree of the original divisor. The remaining expression is the remainder. Thus, your polynomial fraction can be expressed as the quotient plus the remainder over the divisor.