Question

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

Short answer

Quotient: \( x^2 - \frac{2}{3}x - \frac{1}{9} \). Remainder: \( \frac{16}{9}x - \frac{17}{9} \).

Step-by-step solution

Set Up the Division

Write the dividend and divisor in polynomial long division format.Dividend: \( 3x^4 - x^3 + x - 2 \)Divisor: \( 3x^2 + x + 1 \)

Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.\[ \frac{3x^4}{3x^2} = x^2 \] So, the first term of the quotient is \( x^2 \).

Multiply and Subtract

Multiply the entire divisor \( 3x^2 + x + 1 \) by \( x^2 \) and subtract the result from the original dividend. \[ (3x^2 + x + 1) \times x^2 = 3x^4 + x^3 + x^2 \] \[ 3x^4 - x^3 + x - 2 - (3x^4 + x^3 + x^2) = -2x^3 - x^2 + x - 2 \] Update the new polynomial to: \( -2x^3 - x^2 + x - 2 \).

Repeat the Process

Divide \( -2x^3 \) by \( 3x^2 \) to get the next term of the quotient. \[ \frac{-2x^3}{3x^2} = -\frac{2}{3}x \] So, the next term of the quotient is \( -\frac{2}{3}x \).

Multiply and Subtract Again

Multiply the divisor \( 3x^2 + x + 1 \) by \( -\frac{2}{3}x \) and subtract the result from the current polynomial.\[ (3x^2 + x + 1) \times -\frac{2}{3}x = -2x^3 - \frac{2}{3}x^2 - \frac{2}{3}x \] \[ -2x^3 - x^2 + x - 2 - (-2x^3 - \frac{2}{3}x^2 - \frac{2}{3}x) = -\frac{1}{3}x^2 + \frac{5}{3}x - 2 \] Update the new polynomial to: \( -\frac{1}{3}x^2 + \frac{5}{3}x - 2 \).

Find the Final Quotient Term

Divide \( -\frac{1}{3}x^2 \) by \( 3x^2 \) to get the final term of the quotient.\[ \frac{-\frac{1}{3}x^2}{3x^2} = -\frac{1}{9} \] So, the final term of the quotient is \(-\frac{1}{9}\).

Multiply and Subtract One Last Time

Multiply the divisor \( 3x^2 + x + 1 \) by \( -\frac{1}{9} \) and subtract the result from the current polynomial.\[ (3x^2 + x + 1) \times -\frac{1}{9} = -\frac{1}{3}x^2 - \frac{1}{9}x - \frac{1}{9} \] \[ -\frac{1}{3}x^2 + \frac{5}{3}x - 2 - (-\frac{1}{3}x^2 - \frac{1}{9}x - \frac{1}{9}) = \frac{16}{9}x - \frac{17}{9} \] Update the new polynomial to: \( \frac{16}{9}x - \frac{17}{9} \).

Verify the Result

The quotient is \( x^2 - \frac{2}{3}x - \frac{1}{9} \) and the remainder is \( \frac{16}{9}x - \frac{17}{9} \). Verify this by checking that:\[ (3x^2 + x + 1) \cdot (x^2 - \frac{2}{3}x - \frac{1}{9}) + (\frac{16}{9}x - \frac{17}{9}) = 3x^4 - x^3 + x - 2 \]Since this holds true, the solution is correct.

Key concepts

Dividend

The dividend is the polynomial we are dividing. In our example, the dividend is \(3x^4 - x^3 + x - 2\). It represents the numerator in the division operation. Understanding the structure of the dividend is crucial, as it guides the entire division process. Here, the dividend has four terms, with the highest degree of 4. Recognizing the leading term \(3x^4\) and its coefficients sets the foundation for accurate division.

Divisor

The divisor is the polynomial by which we are dividing the dividend. In this case, it is \(3x^2 + x + 1\). The divisor's leading term is particularly important for the division process. The leading term is \(3x^2\) in our divisor, and it plays a key role in determining each term of the quotient. The polynomial structure of the divisor should be clearly understood to perform accurate multiplicative steps during the division.

Quotient

The quotient is the result obtained from dividing the dividend by the divisor. It represents how many times the divisor fits into the dividend. In our exercise, the quotient is constructed step by step:

• First term: \(x^2\)
• Second term: \(-\frac{2}{3}x\)
• Final term: \(-\frac{1}{9}\)

So, the complete quotient is \(x^2 - \frac{2}{3}x - \frac{1}{9}\). Each term is derived by dividing the leading term of the current polynomial (during each step) by the leading term of the divisor.

Remainder

The remainder is what is left after the division process is complete and the divisor can no longer fit into the current polynomial. In this example, the final remainder is \( \frac{16}{9}x - \frac{17}{9} \). The remainder is the lower degree polynomial that remains when the highest possible terms of the divisor have been used to divide the dividend. Checking the correctness of the division involves verifying that:

\((\text{Quotient} \cdot \text{Divisor}) + \text{Remainder} = \text{Dividend}\)
Substituting in our values, we confirm:

\( (3x^2 + x + 1) \cdot (x^2 - \frac{2}{3}x - \frac{1}{9}) + (\frac{16}{9}x - \frac{17}{9}) = 3x^4 - x^3 + x - 2 \)
Since this equation holds true, our quotient and remainder are correct.