Question

Find the quotient and remainder when \(x^{3}+3 x^{2}-6\) is divided by \(x+2\)

Short answer

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

Step-by-step solution

- Set up the Division

The polynomial to be divided is \(x^{3} + 3x^{2} - 6\), and the divisor is \(x + 2\). We'll use polynomial long division to find the quotient and remainder.

- Divide the Leading Terms

Divide the leading term of the dividend \(x^{3}\) by the leading term of the divisor \(x\). This gives \(x^{2}\).

- Multiply and Subtract

Multiply \(x^{2}\) by \(x + 2\) to get \(x^{3} + 2x^{2}\). Subtract this from \(x^{3} + 3x^{2} - 6\), resulting in \(x^{2} - 6\).

- Repeat the Process

Now divide \(x^{2}\) by \(x\) to get \(x\). Multiply \(x\) by \(x + 2\) to get \(x^{2} + 2x\). Subtract this from \(x^{2} - 6\), resulting in \(-2x - 6\).

- Final Division

Now divide \(-2x\) by \(x\) to get \(-2\). Multiply \(-2\) by \(x + 2\) to get \(-2x - 4\). Subtract this from \(-2x - 6\), resulting in \(-2\).

- Write Down the Quotient and Remainder

The quotient is \(x^{2} + x - 2\) and the remainder is \(-2\).

Key concepts

Quotient

When dividing polynomials, the quotient is the result after performing the polynomial division. In simpler terms, it’s what you get when you divide one polynomial by another without considering the remainder yet.

In the given example, the original polynomial is divided by another polynomial, and through a process similar to long division in arithmetic, we find that the quotient is:

• The leading term of the quotient here is determined first by dividing the leading term of the dividend by the leading term of the divisor.
• We continue the process of dividing, multiplying, and subtracting until we’ve gone through each term in the original polynomial.

By consistently working through each term in the polynomial, the resulting quotient for the given exercise is found to be:
\(x^2 + x - 2\).

Remainder

The remainder in polynomial division is what is left over after the division process has been completed. Think of it like the remainder you get when you perform division with whole numbers; it’s the part that does not evenly divide by the divisor.

In our example:

• The division process stops when the degree of the remainder is less than the degree of the divisor.
• We are left with a polynomial, which in this case is \(-2\).
• This \(-2\) does not further divide by \(x+2\) using the polynomial long division method.

Therefore, in the given problem, after completing the polynomial long division, the remainder is: \(-2\).

Long Division

Polynomial long division is a method similar to long division with numbers, used to divide polynomials. This technique helps to determine both the quotient and the remainder.

For our given problem, here’s a quick look at how polynomial long division works:

• We start by dividing the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply this first term of the quotient by the entire divisor and subtract the result from the original dividend.
• Repeat the process with the resulting polynomial, continuing until the degree of the resulting polynomial is less than the degree of the divisor.

In this exercise, that was:

• Starting with \(x^3 + 3x^2 - 6\) as the dividend
• Divided by \(x+2\), following the process step-by-step to get the quotient: \(x^2 + x - 2\) and the remainder: \(-2\).

Understanding polynomial long division is crucial in simplifying expressions and solving polynomial equations. By mastering this technique, you can handle more complex algebraic problems with ease.