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

Quotient: \(3x^2 - 7x + 15\), Remainder: \(-32\).

Step-by-step solution

Set Up the Division

We need to divide the polynomial \[ 3x^3 - x^2 + x - 2 \] by the binomial \[ x + 2 \] using polynomial long division.

Divide the Leading Terms

Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(x\): \[ \frac{3x^3}{x} = 3x^2 \] This gives the first term of the quotient.

Multiply and Subtract

Multiply \(3x^2\) by the entire divisor \(x + 2\) and subtract from the original polynomial: \[ (3x^3 - x^2 + x - 2) - (3x^2(x + 2)) = (3x^3 - x^2 + x - 2) - (3x^3 + 6x^2) = -7x^2 + x - 2 \]

Repeat the Process

Continue dividing, now dividing \(-7x^2\) by \(x\): \[ \frac{-7x^2}{x} = -7x \] This gives the next term of the quotient. Multiply \(-7x\) by \(x + 2\) and subtract: \[ (-7x^2 + x - 2) - (-7x(x + 2)) = (-7x^2 + x - 2) - (-7x^2 - 14x) = 15x - 2 \]

Divide Again

Divide \(15x\) by \(x\): \[ \frac{15x}{x} = 15 \] This gives the final term of the quotient. Multiply \(15\) by \(x + 2\) and subtract: \[ (15x - 2) - (15(x + 2)) = (15x - 2) - (15x + 30) = -32 \] which is the remainder.

Write the Quotient and Remainder

The computation gives us the quotient \(3x^2 - 7x + 15\) and the remainder \(-32\).

Verify the Result

To verify, use the relationship \( \text{Quotient} \, \times \, \text{Divisor} \, + \, \text{Remainder} \): \[ (3x^2 - 7x + 15)(x + 2) - 32 = 3x^3 - x^2 + x - 2 \] Expand and simplify the left-hand side to confirm it equals the original dividend.

Key concepts

Dividend

In polynomial division, the dividend is the polynomial that you are dividing. It's like the numerator in a fraction. For our example, the dividend is the polynomial 3x^3 - x^2 + x - 2. It's the larger polynomial we are trying to break down through the division process.
The steps in the division process start with this polynomial, guiding us to find the quotient and the remainder.
Understanding the dividend helps in setting up the problem and conducting the operations accurately.

Divisor

The divisor is the polynomial that you divide the dividend by. In our example, the divisor is x + 2. It's similar to the denominator in a fraction.
In polynomial long division, you repeatedly divide the highest degree terms of the dividend by the highest degree term of the divisor.
This process helps break down the dividend into smaller, more manageable parts.

• Step-by-step, we use the divisor to determine each term of the quotient.
• It's essential to align the terms correctly to avoid mistakes during multiplication and subtraction.

Quotient

The quotient is the result you get from the division process, excluding the remainder. In our problem, the quotient turns out to be 3x^2 - 7x + 15.
To find each term of the quotient:

• Divide the leading term of the current dividend by the leading term of the divisor.
• Multiply the entire divisor by this term.
• Subtract this product from the current dividend.

This process is repeated until all terms of the original dividend have been divided.
The quotient shows how the original polynomial can be divided into simpler parts with the help of the divisor.

Remainder

After the division process, what remains is called the remainder. In our example, the remainder is -32.
The remainder is the part of the polynomial that cannot be further divided by the divisor.

• It's the 'left-over' portion after completing the polynomial long division steps.
• The remainder can be zero or a polynomial of lower degree than the divisor.

To make sure our solution is correct, we use the remainder in the verification step. We multiply the quotient by the divisor and add the remainder, which should give us back the original dividend.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division of numbers.
Here are the fundamental steps to perform polynomial long division:

• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat the process with the new dividend formed after subtraction, until the degree of the remaining polynomial is less than the degree of the divisor.

This method helps us systematically break down the dividend and manage the calculations in an organized way. Ultimately, polynomial long division gives us both the quotient and the remainder.