Problem 48 - Intermediate Algebra

Question

Perform each division. Divide \(27 a^{3}-8\) by \(3 a-2\)

Step-by-Step Solution

Answer

The quotient is \(9a^2 + 6a + 4\).

Step 1: Setup the Division

We are given a polynomial division problem where we need to divide \(27a^3 - 8\) by \(3a - 2\). Set up the long division by writing \(27a^3 - 8\) under the division bar and \(3a - 2\) outside the division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(27a^3\) by the leading term of the divisor \(3a\). This gives us \(9a^2\) since \(\frac{27a^3}{3a} = 9a^2\). Place \(9a^2\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(9a^2\) by the divisor \(3a - 2\), resulting in \(27a^3 - 18a^2\). Subtract \(27a^3 - 18a^2\) from \(27a^3 - 8\), which leaves us with \(18a^2 - 8\) as the new dividend.

Step 4: Divide the Next Terms

Divide \(18a^2\) by \(3a\), resulting in \(6a\). Add \(6a\) to the quotient above the division line.

Step 5: Multiply and Subtract Again

Multiply \(6a\) by \(3a - 2\), which gives \(18a^2 - 12a\). Subtract \(18a^2 - 12a\) from \(18a^2 - 8\) to get \(12a - 8\) as the new dividend.

Step 6: Divide Again

Divide \(12a\) by \(3a\), which results in \(4\). Add \(4\) to the quotient line.

Step 7: Final Multiplication and Subtraction

Multiply \(4\) by \(3a - 2\) to get \(12a - 8\). Subtract \(12a - 8\) from \(12a - 8\), which gives a remainder of 0.

Step 8: Write the Final Answer

Since there is no remainder, the quotient is \(9a^2 + 6a + 4\). This is the result of the division.

Key Concepts

Long Division

Long division is a powerful method for dividing polynomials, similar to how you might divide numbers. It's a systematic process that helps in breaking down complex algebraic problems into simpler parts.

• First, you set up the division, placing the dividend (the expression we are dividing) under the division bar.
• The divisor (what we are dividing by) is placed outside to the left.

This setup prepares us for the main steps of polynomial division, which include dividing, multiplying, and subtracting the aligned terms until we work through the entire dividend.

For example, when dividing the expression \(27a^3 - 8\) by \(3a - 2\), we methodically go through each term. This method involves repeating the cycle of division and subtraction until the remainder is either zero or a degree less than the divisor.

Understanding long division in polynomials is crucial, as it lays the foundation for tackling more complex algebraic problems.

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical concepts. They consist of variables, numbers, and arithmetic operations.

• Variables are symbols, often letters, that represent unspecified numbers or values.
• These expressions can range from very simple, involving only one or two operations, to very complicated, involving multiple terms and operations.

In the context of polynomial division, we deal specifically with polynomials, which are expressions involving sums of integer powers of a variable.

The power of algebraic expressions is that they allow mathematicians to generalize problems and find solutions applicable to a wide range of specific situations. By working through the division of \(27a^3 - 8\) by \(3a - 2\), we see how these expressions are manipulated step-by-step to achieve a detailed outcome: the quotient.

Learning to handle algebraic expressions in polynomial division broadens one’s ability to navigate various mathematical landscapes.

Quotient and Remainder

In the process of polynomial division, two important results we obtain are the quotient and the remainder. The quotient is the result of the division, while the remainder is what is left over if the division is not exact.

• For division to reach a complete conclusion, the remainder should ideally be zero.
• If there's no remainder, it means the dividend is perfectly divisible by the divisor.

In our example, dividing \(27a^3 - 8\) by \(3a - 2\), we ultimately found a quotient of \(9a^2 + 6a + 4\) with a remainder of 0.

The interplay between the quotient and remainder highlights the precision of algebraic division. Understanding this concept is pivotal for mastering polynomial division, as it indicates when the division process is complete and validates the solution’s accuracy.

By practicing the assessment of quotients and remainders, students gain sharper analytical skills, key for advanced mathematical reasoning.