Problem 37 - Algebra 1

Question

Divide. Divide \(b^{2}-7 b+4\) by \(b+3\)

Step-by-Step Solution

Answer

The answer to the division is \(b -10 + \frac{34}{b+3}\).

Step 1: Set Up the Division

First, set the division up similar to a long division problem: divide \(b^{2}-7 b+4\) by \(b+3\). You put the \(b^{2}-7 b+4\) under the division symbol and \(b+3\) outside to the left.

Step 2: Division Process - First Round

Divide the first term in the dividend (\(b^{2}\)) by the first term in the divisor (\(b\)). This gives us \(b\), which becomes the first term in our solution. Multiply \(b+3\) by \(b\), which results in \(b^{2}+3b\), and subtract this from \(b^{2}-7 b\). This process leaves us with \( -10b\).

Step 3: Division Process - Second Round

Now, divide the \( -10b\) by the first term in the divisor (\(b\)), resulting in \( -10\). This becomes our second term in the solution. Multiply \(b+3\) by \( -10\), which gives us \(-10b -30\). Subtract this from our remaining terms \(-10b+4\), we get \(34\).

Step 4: Write Down the Final Answer

As the degree of the remaining term (\(34\)) is lower than the degree of the divisor (\(b+3\)), we cannot continue the division. Thus, \(34\) is the remainder. Therefore, the answer to the division is \(b -10 + \frac{34}{b+3}\).

Key Concepts

long division method

The long division method is a systematic technique for dividing polynomials, much like how you would divide numbers. It involves dividing the terms of the polynomials step-by-step. First, compare the highest degree terms of the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by). This long division approach helps simplify complex expressions by breaking down the division process into manageable steps.

Here’s a quick overview of how it works:

• Setting Up the Problem: Write the dividend under the division symbol and the divisor outside to the left. Similar to numerical long division, the structure helps keep the calculation organized.
• Dividing Terms: Start by dividing the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient.
• Multiplying and Subtracting: Multiply the entire divisor by this quotient term and subtract the result from the original dividend. The subtraction reveals the next term that needs to be divided.
• Repeat: Continuously perform these steps (dividing, multiplying, and subtracting) until you reach a remainder with a degree lesser than that of the divisor.

Understanding each step ensures that you accurately perform polynomial division using the long division method.

remainder in algebra

The remainder in algebraic division is the leftover part of the dividend that cannot be divided further by the divisor. When dividing polynomials using long division or synthetic division methods, you may not always end up with a zero remainder. This remainder is an important component and must be considered in the final answer.

To grasp this concept, remember:

• Degree Rule: The remainder will have a lower degree than the divisor. This rule is crucial as it dictates when to stop the division process.
• Final Quotient: The answer from the division is expressed as a quotient with an additional term representing the remainder, written as a fraction (remainder over divisor).
• Verification: To check if your division is correct, you can multiply the divisor by the quotient obtained and add the remainder. The result should equal the original dividend.

These points help ensure that you understand and accurately compute the remainder in polynomial division, facilitating error-free solutions.

polynomial expressions

Polynomial expressions are mathematical expressions involving sums of powers of variables, each term having a coefficient. They form the foundation of algebra and are essential in many real-world applications like physics and engineering.

When handling polynomial expressions, it’s important to be familiar with:

• Structure: Each polynomial is made up of terms. A term consists of a coefficient multiplied by a variable raised to a power. The degree of the polynomial is determined by the highest power of the variable.
• Simplification: Polynomials often need to be simplified by combining like terms, which are terms with the same variable raised to the same power. This process reduces complexity and prepares polynomials for operations like addition, subtraction, multiplication, or division.
• Operations: Understanding how to perform operations on polynomials—such as addition, subtraction, and particularly division—is crucial. Each operation follows its own set of rules, especially division which is more intricate.

Being comfortable with polynomial expressions is vital for comprehending higher-level algebraic concepts and solving complex problems.