Problem 81 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 a^{2} b-3 b^{2}-5 b^{2}-8 q^{2} b-10 a^{2} b-b^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(4a^{2}b - 3b^{2} - 10a^{2}b - 5b^{2} - 8q^{2}b - b^{2}\). Answer: \(-6a^{2}b - 9b^{2} - 8q^{2}b\)

Step 1: Combine Like Terms: Term 1, Term 2, and Term 4 with a^{2}b and Term 3 and Term 6 with b^{2}

First, identify the like terms in the expression. The like terms are those terms which have the same variables with the same powers: \(4a^{2}b\) and \(-10a^{2}b\) are like terms because they share the variable factors \(a^{2}b\). In addition, \(-3b^{2}\) and \(-5b^{2}\) and \(-b^{2}\) are like terms because they share the variable factor \(b^{2}\).

Step 2: Combine the Coefficients of the Terms with the Common Variable Powers

Now that we have identified the like terms, we will combine their coefficients: Term 1 and Term 4: \(4a^{2}b\) and \(-10a^{2}b\) Combine their coefficients: \(4 - 10 = -6\) New combined term: \(-6a^{2}b\) Term 2, Term 3, and Term 6: \(-3b^{2}, -5b^{2}\), and \(-b^{2}\) Combine their coefficients: \(-3 - 5 - 1 = -9\) New combined term: \(-9b^{2}\)

Step 3: Rewrite the Remaining Term and Combine All Simplified Terms

The given expression also has the term \(-8q^{2}b\) that has not been combined with any other terms because there are no like terms for this particular term. Rewrite this term unchanged. Now, combine the simplified terms along with the unchanged term \(-8q^{2}b\): $$ -6a^{2}b - 9b^{2} -8q^{2}b $$ The algebraic expression has now been simplified.

Key Concepts

Like Terms

Understanding like terms is a foundational step in simplifying algebraic expressions. Like terms are those that have the exact same variable part with identical exponents. For example, in the expression given, terms like \(4a^{2}b\) and \(-10a^{2}b\) are considered like terms because they both contain the variable piece \(a^{2}b\). Similarly, terms such as \(-3b^{2}\), \(-5b^{2}\), and \(-b^{2}\) are like terms due to sharing the \(b^{2}\) variable component.
To simplify, first identify all sets of like terms. This step sets the stage for combining them, which makes the expression much more manageable:

• Look for terms with the same variable and same power.
• Group these terms together visually or mentally to make the next step (coefficient combination) easier.
  

This concept is crucial as it ensures we are adding and subtracting only comparable components, maintaining the expression's mathematical integrity.

Coefficient Combination

Once like terms are identified, the next step is combining their coefficients. Coefficients are the numerical part attached to the variables. In the expression \(4a^{2}b - 3b^{2} - 5b^{2} - 8q^{2}b - 10a^{2}b - b^{2}\), let's focus on the coefficients:

• For \(4a^{2}b\) and \(-10a^{2}b\), the coefficients 4 and -10 are combined: \(4 - 10 = -6\)
• For \(-3b^{2}\), \(-5b^{2}\), and \(-b^{2}\), the coefficients -3, -5, and -1 are summed: \(-3 - 5 - 1 = -9\)

By focusing on the coefficients, we simplify the process. This becomes a straightforward arithmetic operation, allowing us to rewrite the expression with fewer terms. Combining coefficients correctly is important to maintain the expression's value.
Keep the variable part the same while working only on the numerical values. This way, the variables align correctly, leading to a properly simplified outcome.

Expression Simplification

After combining like terms and their coefficients, the final step is to rewrite the simplified expression. This involves bringing together all the newly formed terms into one streamlined form. In the given problem, you arrive at:
\[-6a^{2}b - 9b^{2} -8q^{2}b\]
This expression is much simpler and easier to interpret compared to the original. The simplification process is crucial in algebra because it allows you to solve equations, perform further calculations, and understand mathematical relationships more clearly.

• Ensure every term has been considered; nothing should be left out unless it’s unmatched, like \(-8q^{2}b\) here.
• Simplified expressions are more efficient for solving and manipulating within broader mathematical problems.
  

Understanding these steps and applying them helps in mastering algebraic manipulation. With practice, simplifying expressions becomes intuitive, making algebra less intimidating and more approachable.