Problem 65 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{6 a^{2}+5 a-6}{2 a+3} $$

Step-by-Step Solution

Answer

The result of the division is \(3a - 2\).

Step 1: Set Up the Division

We are tasked with dividing the polynomial \(6a^2 + 5a - 6\) by \(2a + 3\). Start by writing this as a long division problem, with \(2a+3\) as the divisor and \(6a^2 + 5a - 6\) as the dividend.

Step 2: Divide the First Term

Divide the first term of the dividend, \(6a^2\), by the first term of the divisor, \(2a\). This gives \(3a\), which will be the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(2a + 3\) by the term obtained in Step 2, which is \(3a\). This gives \(6a^2 + 9a\). Subtract this result from the original dividend \(6a^2 + 5a - 6\), resulting in a new polynomial \(-4a - 6\).

Step 4: Divide the New First Term

Divide the new first term \(-4a\) by \(2a\), which results in \(-2\). This is the next term in the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-2\) by the entire divisor \(2a + 3\) to get \(-4a - 6\). Subtracting this from \(-4a - 6\) results in \(0\).

Step 6: Write the Final Answer

Since we have reached a remainder of 0, the division is complete. The quotient of the division \(\frac{6a^2 + 5a - 6}{2a + 3}\) is \(3a - 2\).

Key Concepts

Long Division

Long division in algebra is similar to the arithmetic long division, and it is used when dividing polynomials. Think of it as a process where we repeatedly divide, multiply, and subtract to eliminate terms from the dividend, step by step, until no terms remain.
This approach helps break down complex expressions into simpler parts to find the quotient and remainder.
Here are a few key steps in polynomial long division:

• Start by dividing the first term of the dividend by the first term of the divisor. Write this result as part of the quotient.
• Then, multiply the entire divisor by the term you just found, and subtract this product from the original dividend.
• The subtraction results in a new polynomial. Repeat the process using this new polynomial as the dividend until you cannot divide anymore, meaning either you reach a remainder or zero.

Using long division makes understanding polynomial expressions and their relationships clearer, especially when dealing with algebraic expressions.

Divisor

In the context of polynomial division, the divisor is the polynomial that you are dividing by. In our exercise, the divisor is the term \(2a + 3\).
Knowing the divisor's role is crucial because it is used to determine each subsequent term in the quotient.
To efficiently perform polynomial division, follow these tips:

• Identify the leading term of the divisor, which will be vital for determining how to simplify the dividend.
• Divide only using the leading term of the divisor to simplify the process.
• Ensure all terms in the divisor are accounted for when performing multiplication and subtraction to maintain accuracy.

By understanding the divisor's part in polynomial division, you can more easily see how each term in the dividend is affected in the division process.

Dividend

The dividend is the polynomial you wish to divide. In our example, \(6a^2 + 5a - 6\) is the dividend.
The dividend contains all the terms that need to be paired down through the division process to find the quotient.
Key pointers for managing the dividend during polynomial division:

• Write the polynomial in descending powers of the variable to maintain order.
• Perform operations starting with the leading term to simplify the polynomial effectively.
• Systematically match terms with the divisor to ensure no terms are left over except for the remainder or zero.

Understanding the structure and order of the dividend allows a seamless transitions through the problem-solving process.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the essence of algebra and are manipulated in various ways to solve mathematical problems.
In this division problem, both the divisor and dividend are algebraic expressions.
Important aspects of handling such expressions include:

• Recognizing the type of expression, whether it's a polynomial, a rational expression, or another form.
• Simplifying expressions by combining like terms or factoring where necessary to make them more manageable.
• Using division, multiplication, and other operations to explore relationships and solve for unknowns.

Understanding algebraic expressions and their manipulation is central to mastering algebraic concepts, including polynomial division and beyond.