Question

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ 2 x^{4}-3 x^{3}+x+1 \text { divided by } 2 x^{2}+x+1 $$

Short answer

Quotient: \( x^2 - 2x + \frac{1}{2} \), Remainder: \( \frac{5}{2}x + \frac{1}{2} \)

Step-by-step solution

Set Up Polynomial Long Division

Write the dividend, which is the polynomial to be divided: \[ 2x^4 - 3x^3 + x + 1 \]Write the divisor, which is the polynomial we are dividing by: \[ 2x^2 + x + 1 \]

Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{2x^4}{2x^2} = x^2 \]This is the first term of the quotient.

Multiply and Subtract

Multiply \( x^2 \) by the entire divisor \( 2x^2 + x + 1 \) and subtract the result from the original dividend:\[ (2x^4 - 3x^3 + x + 1) - (x^2(2x^2 + x + 1)) \]Perform the operation: \[ 2x^4 - 3x^3 + x + 1 - (2x^4 + x^3 + x^2) \]This simplifies to: \[ -4x^3 - x^2 + x + 1 \]

Repeat the Process

Repeat the steps: Divide the new leading term \( -4x^3 \) by the leading term of the divisor. \[ \frac{-4x^3}{2x^2} = -2x \]Subtract the result: \[ -4x^3 - x^2 + x + 1 - (-2x)(2x^2 + x + 1) \]Perform the operation: \[ -4x^3 - x^2 + x + 1 - (-4x^3 - 2x^2 - 2x) \]This simplifies to: \[ x^2 + 3x + 1 \]

Divide the Leading Terms Again

Divide \( x^2 \) by the leading term of the divisor: \[ \frac{x^2}{2x^2} = \frac{1}{2} \]Subtract the result: \[ x^2 + 3x + 1 - \frac{1}{2}(2x^2 + x + 1) \]Perform the operation: \[ x^2 + 3x + 1 - (x^2 + \frac{1}{2}x + \frac{1}{2}) \]This simplifies to: \[ \frac{5}{2}x + \frac{1}{2} \]

Identify Quotient and Remainder

The quotient is: \[ x^2 - 2x + \frac{1}{2} \]The remainder is: \[ \frac{5}{2}x + \frac{1}{2} \]

Check Your Work

Multiply the quotient by the divisor and add the remainder to check if it gives the original dividend:\[ (x^2 - 2x + \frac{1}{2})(2x^2 + x + 1) + \frac{5}{2}x + \frac{1}{2} \]Perform the multiplication and addition to verify it matches the dividend \( 2x^4 - 3x^3 + x + 1 \).

Key concepts

Understanding the Dividend

In polynomial long division, the **dividend** is the polynomial you want to divide. Think of it as the 'numerator' in regular fraction division. It is the polynomial we break down into smaller parts. In our example, the dividend is:
\[ 2x^4 - 3x^3 + x + 1 \] The goal is to simplify the dividend using the divisor through a sequence of steps until we either fully simplify the polynomial or identify a remainder. Identifying and setting up your dividend correctly is the first crucial step in solving these types of problems.

Examining the Divisor

The **divisor** in polynomial long division is the polynomial you are dividing by, similar to the 'denominator' in fraction division. For the given problem, our divisor is:
\[ 2x^2 + x + 1 \] It is essential to correctly identify and write down the divisor because we will use it repeatedly in the division process. By dividing the leading terms of the dividend by the leading terms of the divisor and performing subsequent operations, we continue to break down the dividend in each step.

Finding the Quotient

The **quotient** is the result you get when you divide the dividend by the divisor. It is the 'answer' to the division problem, excluding any remainder. In polynomial division, the quotient can be another polynomial. In the example problem, after several steps of dividing and subtracting, we find our quotient step-by-step. The final quotient is:

• First part: \[ x^2 \]
• Second part: \[ -2x \]
• Third part: \[ \frac{1}{2} \]

So, the complete quotient is: \[ x^2 - 2x + \frac{1}{2} \]

Determining the Remainder

After dividing the dividend by the divisor and getting the quotient, whatever is left over is known as the **remainder**. In polynomial division, this remainder can also be a polynomial of lower degree than the divisor. In our example, after performing all the steps of polynomial division, we are left with:
\[ \frac{5}{2}x + \frac{1}{2} \]This is our remainder. If there was no remainder, the division would be exact. To verify the solution, you can multiply the quotient by the divisor and add the remainder. If done correctly, this should equal the original dividend. In this case: \[ (x^2 - 2x + \frac{1}{2})(2x^2 + x + 1) + \frac{5}{2}x + \frac{1}{2} \]will match \[ 2x^4 - 3x^3 + x + 1 \], confirming our solution.