Question

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ 3 x^{5}-x^{2}+x-2 \text { divided by } 3 x^{3}-1 $$

Short answer

Quotient: \(x^2\), Remainder: \(x - 2\)

Step-by-step solution

- Set up the long division

Write the dividend and the divisor in long division format. The dividend is the polynomial \(3x^5 - x^2 + x - 2\), and the divisor is \(3x^3 - 1\).

- Divide the first term

Divide the highest degree term of the dividend \(3x^5\) by the highest degree term of the divisor \(3x^3\). This gives you \(x^2\).

- Multiply and subtract

Multiply \(x^2\) by the entire divisor \(3x^3 - 1\) to get \(3x^5 - x^2\). Subtract this product from the dividend to get a new polynomial: \(0 + 0 + x - 2\).

- Bring down the next term

There are no remaining terms that have a degree equal to or greater than that of the divisor. Thus, the quotient so far is \(x^2\) and the remainder is the polynomial \(x - 2\).

- Verify the solution

To verify, multiply the quotient \(x^2\) and the divisor \(3x^3 - 1\) to obtain \(3x^5 - x^2\). Add the remainder \(x - 2\) to this product to get the original dividend \(3x^5 - x^2 + x - 2\). This verifies that the solution is correct.

Key concepts

Dividend

In polynomial long division, the term 'dividend' refers to the polynomial you are dividing. Think of it as the number inside the division symbol in traditional arithmetic division. In our exercise, the dividend is the polynomial: \[3x^5 - x^2 + x - 2\]

• The dividend is written first.
• It is placed inside the long division bar.
• This polynomial is divided by another polynomial, the divisor.

Understanding the structure helps in setting up the problem correctly for division.

Divisor

The 'divisor' in polynomial long division is the polynomial by which you are dividing the dividend. It's similar to the number outside the division bar in arithmetic division. For our problem, the divisor is: \[3x^3 - 1\]

• The divisor is placed outside the long division bar.
• We divide the leading term of the dividend by the leading term of the divisor.
• This helps to progressively reduce the polynomial until we are left with a simpler form.

Make sure to carefully align the terms by their degree for easier calculations.

Quotient

The 'quotient' in polynomial division is the result you get when you divide the dividend by the divisor. It's the series of terms you build as you perform the division step by step. In our example, the quotient starts forming with the first term:\[x^2\]

• It starts with dividing the highest degree term of the dividend by the highest degree term of the divisor.
• Next, this quotient term is multiplied by the entire divisor, and the result is subtracted from the dividend.
• Any remaining polynomial after the process becomes our new dividend for the subsequent steps.

By repeating these steps, you build the complete quotient. Here, the quotient is: \[x^2\]

Remainder

In polynomial division, the 'remainder' is what's left after the division process. This is similar to the leftover part in standard arithmetic division. In our specific problem, we get the remainder after performing the necessary subtraction:\[x - 2\]

• The remainder has a lower degree than the divisor.
• If the remainder is zero, it means the division is exact.
• If not, it shows how much is left after dividing as far as possible.

In this example, since there are no like terms left to divide, the remainder is simply carried over.