Question

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

Short answer

The quotient is \(3x - 1\) and the remainder is \(x - 2\).

Step-by-step solution

Set Up the Division

Write the polynomial long division setup with the dividend and divisor. Dividend: \(3x^3 - x^2 + x - 2\) Divisor: \(x^2\)

Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor: \( \frac{3x^3}{x^2} = 3x\). The first term of the quotient is \(3x\).

Multiply and Subtract

Multiply the entire divisor by the first term of the quotient: \(3x \times x^2 = 3x^3\). Subtract this result from the dividend: \((3x^3 - x^2 + x - 2) - 3x^3 = -x^2 + x - 2\).

Divide the New Leading Term

Divide the new leading term of the current polynomial by the leading term of the divisor: \( \frac{-x^2}{x^2} = -1\). This is the second term of the quotient.

Multiply and Subtract Again

Multiply the entire divisor by the new term of the quotient: \(-1 \times x^2 = -x^2\). Subtract this from the current polynomial: \((-x^2 + x - 2) - (-x^2) = x - 2\).

Identify the Quotient and Remainder

After completing the division, the quotient is \(3x - 1\) and the remainder is \(x - 2\).

Verify the Result

Verify by checking if (Quotient \( \cdot \) Divisor \( + \) Remainder) equals the original dividend: \((3x - 1) \cdot x^2 + (x - 2) = 3x^3 - x^2 + x - 2\). Since it matches, our result is verified.

Key concepts

Long Division

Polynomials can be divided using a method similar to long division in arithmetic. This process helps us break down complex polynomials into simpler terms. Start by setting up your division: place the dividend (the polynomial to be divided) inside the long division symbol and the divisor (the polynomial you are dividing by) outside.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, and the remainder is what is left after the division is complete. Begin by dividing the leading term of the dividend by the leading term of the divisor. This quotient term will be used to multiply the entire divisor, and then subtract that product from the dividend. Repeat this process until you can no longer divide because the degree of the remaining polynomial is less than that of the divisor. In this example, we ended up with a quotient of \(3x - 1\) and a remainder of \(x - 2\).

Polynomial Verification

After finding the quotient and the remainder, it's important to verify the result. This ensures that your division was performed correctly. To verify, multiply the quotient by the divisor and then add the remainder. If it matches the original dividend, then the division is correct. In this case, when you multiply \(3x - 1\) by \(x^2\) and then add \(x - 2\), you should get back to \(3x^3 - x^2 + x - 2\), confirming that your division was accurate.