Question

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

Short answer

Quotient: \(x^2 + ax + a^2\); Remainder: 0.

Step-by-step solution

- Understand the Problem

We need to divide the polynomial \(x^3 - a^3\) by \(x - a\) and find both the quotient and the remainder. We will verify the result by using the relationship given: Quotient \(\cdot\) Divisor \( + \) Remainder \( = \) Dividend.

- Use Polynomial Long Division or Synthetic Division

Since \(x^3 - a^3\) can be factored using the difference of cubes formula: \(x^3 - a^3 = (x - a)(x^2 + ax + a^2)\), we can see that the divisor \(x - a\) will cancel out in the division process.

- Identify the Quotient and Remainder

From step 2, the quotient is \(x^2 + ax + a^2\) and the remainder is \(0\) because \(x - a\) divides evenly into \(x^3 - a^3\).

- Verify the Result

Multiplying the quotient by the divisor and then adding the remainder should give us back the original dividend: \[ (x^2 + ax + a^2) \cdot (x - a) + 0 = x^3 - a^3. \]Since the right-hand side is the original dividend, our solution is verified.

Key concepts

Difference of Cubes

When we look at polynomials like \(x^3 - a^3\), we need to recognize special patterns to simplify them. One such pattern is the difference of cubes. This concept comes in handy for breaking down complex polynomials.

The difference of cubes follows this formula: \[x^3 - a^3 = (x - a)(x^2 + ax + a^2).\]

This formula is extremely useful because it allows us to factor a cubic polynomial into a product of a linear term \(x - a\) and a quadratic term \(x^2 + ax + a^2\). By factoring, we can easily divide these polynomials and find the quotient and remainder. In our exercise, this helped simplify the division process significantly.

Synthetic Division

Synthetic division is a simplified way to divide polynomials, especially when the divisor is in the form \(x - c\). This method can save time compared to the traditional long division method.

Here’s how it works:

• Write down the coefficients of the polynomial being divided.
• Use the root of the divisor term \(x - a\), which here is \(a\).
• Bring down the first coefficient and then multiply it by \(a\).
• Add the result to the next coefficient horizontally and repeat the process for all coefficients.

This technique works efficiently for lower-degree polynomials and helps in quickly finding the quotient and remainder. In our example, however, factoring played a key role, and synthetic division helped confirm our results.

Verifying Division Results

It's always important to verify the results after performing polynomial division. This not only checks your work but also ensures that the quotient and the remainder are correct.

For verifying our division, we use the formula:
Quotient \( \cdot \) Divisor \(+\text{Remainder} = \text{Dividend}.\)
In our exercise, we found that the quotient is \(x^2 + ax + a^2\) and the remainder is \(0\). Using the verification formula:
\((x^2 + ax + a^2) \cdot (x - a) + 0 \ = x^3 - a^3\)

As we multiply and simplify, we see it matches the original dividend \( x^3 - a^3 \) confirming our result is correct.