Problem 47 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{a^{3}-8}{a-2} $$

Step-by-Step Solution

Answer

Answer: The result of the polynomial division is $$a^2 + 2a + 4$$.

Step 1: Set up the long division

To perform long division on $$\frac{a^3 - 8}{a-2},$$ we first set up the long division by placing the divisor $$a-2$$ on the left and the dividend $$a^3 - 8$$ on the right: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{a^2} & +2a & +4 \\ \cline{2-5} a-2 & a^3 & & & -8 \\ \cline{2-2} \multicolumn{2}{r}{a^3} & -2a^2& & \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & +2a^2& & \\ \multicolumn{2}{r}{\underline{\phantom{xx}}} & +2a^2& -4a& \\ \cline{3-4} \multicolumn{2}{r}{\underline{\phantom{xx}}} & & +4a & -8 \\ \multicolumn{2}{r}{\underline{\phantom{xx}}} & & +4a & -8 \\ \cline{4-5} \multicolumn{2}{r}{\underline{\phantom{xx}}} & & & 0 \\ \end{array} $$

Step 2: Read the result

From the long division, we can now read the result. The division results in a quotient of $$a^2 + 2a + 4$$ with a remainder of $$0$$. So, the final answer is: $$\frac{a^3 - 8}{a-2} = a^2 + 2a + 4$$

Key Concepts

Polynomial Division

Polynomial division is a method to divide one polynomial by another. It's like the long division we used in arithmetic, but with variables involved. This technique is particularly useful when working with higher degree polynomials.

When dividing polynomials, we place the divisor (the polynomial we are dividing by) on the left and the dividend (the polynomial we want to divide) on the right. The goal is to find how many times the divisor fits into the dividend, just like when you divide numbers.

In the exercise, we divided \( a^3 - 8 \) by \( a - 2 \). We began by aligning terms based on their powers, focusing on one term at a time to systematically reduce the polynomial until we are left with a simpler expression.

Quotient and Remainder

The result of polynomial division consists of a quotient and sometimes a remainder. The quotient is the answer to the division, while the remainder is what is left over after the division is complete.

In our problem, after completing the algebraic long division of \( a^3 - 8 \) by \( a - 2 \), we found that the quotient was \( a^2 + 2a + 4 \) and the remainder was 0. A remainder of zero indicates that \( a^3 - 8 \) is exactly divisible by \( a - 2 \), meaning it can be expressed as a product of \( a - 2 \) and \( a^2 + 2a + 4 \).

This concept is similar to finding that 18 divided by 3 gives a quotient of 6 with no remainder, showing that 3 is a factor of 18.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. These operations allow us to simplify complex expressions and solve polynomial equations.

When dividing polynomials, each operation contributes to simplifying the expression. For instance, during our division task, we performed multiple subtractions to eliminate terms step by step. Each subtraction brought us closer to the final quotient.

These operations are crucial in calculus, algebra, and many applied mathematical fields. Mastering them allows us to handle large-scale computations and solve real-world problems efficiently. In our example, the ability to divide polynomials helped break down \( a^3 - 8 \) into simpler, more manageable parts.