Problem 26 - Algebra 2

Question

Simplify. $$ \left(2 b^{3}+b^{2}-2 b+3\right)(b+1)^{-1} $$

Step-by-Step Solution

Answer

\(2b^2 - b - 1 + \frac{4}{b+1}\).

Step 1: Identify the Expression

The expression given is \( \left(2b^3 + b^2 - 2b + 3\right)(b + 1)^{-1} \). We need to simplify this by removing the fraction.

Step 2: Simplify the Denominator Expression

The expression \((b + 1)^{-1}\) can be simplified by realizing it indicates division by \((b + 1)\). Therefore, we are dividing \(2b^3 + b^2 - 2b + 3\) by \((b + 1)\).

Step 3: Set Up Polynomial Long Division

We set up polynomial long division where \(2b^3 + b^2 - 2b + 3\) is divided by \(b + 1\). The divisor is \(b + 1\) and the dividend is \(2b^3 + b^2 - 2b + 3\).

Step 4: Begin Division Process

Divide the first term of the dividend, \(2b^3\), by the first term of the divisor, \(b\), to get \(2b^2\). Multiply \(2b^2\) by \(b + 1\), resulting in \(2b^3 + 2b^2\), and subtract from the dividend.

Step 5: Secondary Division Step

Subtract \(2b^3 + 2b^2\) from the initial dividend to get \(-b^2 - 2b + 3\). Divide \(-b^2\) by \(b\) to get \(-b\). Multiply \(-b\) by \(b + 1\), obtaining \(-b^2 - b\), and subtract again.

Step 6: Final Steps of Division

Subtracting yields \(-b+3\). Now divide \(-b\) by \(b\) to get \(-1\). Multiply \(-1\) by \(b + 1\), giving \(-b - 1\), and perform the final subtraction from \(-b+3\) to get a remainder of \(4\).

Step 7: Interpret the Result

The division result is \(2b^2 - b - 1\) with a remainder of \(4\). Therefore, the expression simplifies to \(2b^2 - b - 1 + \frac{4}{b+1}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making them easier to work with, while maintaining their equality. It's a bit like tidying up a messy room—you want everything neat and in its appropriate place, without losing any items. When dealing with algebraic expressions, our goal is to transform them into a form that's more manageable and easy to understand. This often involves:

• Combining like terms
• Canceling out terms where possible
• Removing unnecessary parentheses

In the given exercise, simplification starts by identifying the structure: We have (2b^3 + b^2 - 2b + 3)(b+1)^{-1}, which can be interpreted as dividing 2b^3 + b^2 - 2b + 3 by b+1.
The aim is to remove the fraction aspect presented by (b+1)^{-1} and adjust the expression into a cleaner polynomial form through division.

Polynomial Division

Polynomial division works much like long division you might have done with regular numbers, but instead, we use expressions with variables. This technique is crucial for simplifying complex polynomial expressions and is particularly helpful as it allows you to divide a larger polynomial, the dividend, by a smaller one, the divisor.

• The first task is to arrange both the dividend and divisor in descending order of power.
• We divide the first term of the dividend by the first term of the divisor. This gives us the leading term of the quotient.
• The next step involves multiplying this term by the entire divisor and subtracting the result from the original polynomial.
• Repeat the process until the remaining expression's degree is less than the divisor's.

The exercise involves dividing 2b^3 + b^2 - 2b + 3 by b+1. Using polynomial long division, we find the simplified result as 2b^2 - b - 1 with a remainder of 4.

Remainder Theorem

The remainder theorem provides a quick way to evaluate the result of a polynomial division, offering a significant time-saving advantage if you're interested only in specific results such as the remainder. According to this theorem, if you divide a polynomial P(x) by (x - c), where x-c is a linear factor, the remainder of this division is P(c). This concept works beautifully for understanding division's outcome without going through all steps.
In this exercise, after performing polynomial long division with b+1 (as an equivalent to (b-(-1))), the remainder obtained is 4. This makes sense because, if we replace b with -1 in the original polynomial (2b^3 + b^2 - 2b + 3), we find that the result equals 4. This confirms the accuracy of the division process by validating our final remainder.