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Chapter 1: Problem 11

$$\text { Simplify the given algebraic expressions.}$$ $$a^{2} b-a^{2} b^{2}-2 a^{2} b$$

Short Answer

Expert verified
The simplified expression is \(-a^{2}b(1 + b)\).

Step by step solution

01

Combine Like Terms

To simplify the expression, start by combining like terms. The given expression is \(a^{2}b - a^{2}b^{2} - 2a^{2}b\). The like terms are \(a^{2}b\) and \(-2a^{2}b\). Add these two terms together: \(a^{2}b - 2a^{2}b = -a^{2}b\).
02

Simplify the Expression

With the like terms combined, we have the new expression \(-a^{2}b - a^{2}b^{2}\). Notice that both terms share a common factor, \(-a^{2}b\). We can factor \(-a^{2}b\) out of the expression: \(-a^{2}b(1 + b)\).
03

Final Simplified Form

The expression \(-a^{2}b(1 + b)\) is already in its simplest form, as it cannot be simplified further. Therefore, this is the final simplified version of the given algebraic expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When simplifying algebraic expressions, the first step is often to identify and combine like terms. Like terms are terms that have identical variable parts with the same exponents. For example, in the expression \( a^2b - a^2b^2 - 2a^2b \), the terms \( a^2b \) and \(-2a^2b \) are like terms because they both contain the same variables with the same exponents.
  • To combine them, you simply add or subtract the coefficients in front of the variable expressions.
  • In this case, you have 1 \( a^2b \) and -2 \( a^2b \), which combine to give \( -a^2b \).
By combining like terms, you reduce the complexity of the algebraic expression, making it easier to handle in any subsequent operations.
Factoring Algebraic Expressions
Factoring is a crucial technique for simplifying expressions, especially when you want to identify common factors in different parts of the expression. Factoring involves finding a term that is common to all parts of the expression and "factoring it out" to simplify the expression into a product of simpler expressions. In the expression \( -a^2b - a^2b^2 \), after combining like terms, notice that both terms contain a common factor of \( -a^2b \).
  • Factoring \( -a^2b \) out of the expression, you are left with a simpler expression: \(-a^2b(1 + b)\).
  • This transformation helps you see the structure of the expression clearly and sometimes makes further operations much simpler.
Remember, identifying common factors can significantly simplify complex algebraic expressions.
Polynomial Simplification
Polynomial simplification is an overarching term for combining like terms and factoring. The goal is to reduce a polynomial to its simplest form. A polynomial is in its simplest form when it is expressed as a sum or a difference with no like terms and all common factors factored out. For the expression \( -a^2b(1 + b) \), we have reached a simplified form through both combining like terms and factoring.
  • Simplifying polynomials involves first making sure that all like terms are combined.
  • Then, factoring out common terms where possible ensures the expression is made up of irreducible factors.
  • The final form is tidy and typically the most useful for further calculations in solving equations or graphing.
Mastering these steps in polynomial simplification paves the way for tackling more complex algebraic challenges efficiently.

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Most popular questions from this chapter

Solve the given problems. A car costs \(\$ 38,000\) new and is worth \(\$ 24,000\) 2 years later. The annual rate of depreciation is found by evaluating \(100(1-\sqrt{V / C})\) where \(C\) is the cost and \(V\) is the value after 2 years. At what rate did the car depreciate? (100 and 1 are exact.)

Perform the indicated operations. The first number is approximate, and the second number is exact. $$0.9788+14.9$$

Solve the given problems. For what values of \(x\) is (a) \(x>\sqrt{x},\) (b) \(x=\sqrt{x}\), and (c) \(x<\sqrt{x} ?\)

Find the value of each square root by use of a calculator. Each number is approximate. (a) \(\sqrt{0.0429^{2}-0.0183^{2}}\) (b) \(\sqrt{0.0429^{2}}-\sqrt{0.0183^{2}}\)

In finding the rate \(v\) (in \(\mathrm{km} / \mathrm{h}\) ) at which a polluted stream is flowing, the equation \(15(5.5+v)=24(5.5-v)\) is used. Find \(v.\)
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