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Chapter 4: Problem 71

For the following problems, simplify each of the algebraic expressions. $$ 4 a-a(a+5) $$

Short Answer

Expert verified
Question: Simplify the given algebraic expression: $$4a - a(a+5).$$ Answer: The simplified expression is $$-a^2 - a.$$

Step by step solution

01

Distribute the Multiplication

Begin by applying the distributive property of multiplication to the given expression: multiply $$a$$ with both $$a$$ and $$5$$. $$ 4a - a(a+5) = 4a - a^2 - 5a. $$
02

Combine Like Terms

Combine the like terms, which are the terms with the variable $$a$$: $$ 4a - a^2 - 5a = -a^2 + (4a - 5a). $$ Subtract $$5a$$ from $$4a$$: $$ -a^2 + (4a - 5a) = -a^2 - a. $$
03

Final Simplified Expression

The final simplified expression is: $$ -a^2 - a. $$

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle in algebra that allows you to multiply a single term by each term within a set of parentheses. It simplifies expressions that involve multiplication across sums or differences. In this exercise, you start with the expression:
\[4a - a(a + 5)\]
To apply the distributive property, you multiply \(a\) by each term inside the parentheses \((a + 5)\). This gives you:
  • \(a \times a = a^2\)
  • \(a \times 5 = 5a\)
Thus, the expression transforms into:
\[4a - a^2 - 5a\]
This property essentially removes the parentheses by distributing the multiplication across the terms inside.
Combining Like Terms
Combining like terms is the process of simplifying expressions by adding or subtracting terms that contain the same variables raised to the same power. This step helps to condense expressions into their simplest form.
In the expression derived from our distribution step:
\[4a - a^2 - 5a\]
The like terms are those that include the variable \(a\). Here, you have:
  • \(4a\), which is a positive term
  • \(-5a\), which is a negative term
By subtracting \(5a\) from \(4a\), you simplify the expression to:
\[-a^2 - a\]
This step is crucial as it consolidates terms to ensure the expression is as simple as possible.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It represents a specific value based on the variables and numbers involved.
For instance, the expression from our exercise:
\[4a - a(a + 5)\]
is an algebraic expression involving the variable \(a\). Each component of this expression鈥擻(4a\), \(-a(a+5)\)鈥攑lays a role in the overall calculation. Understanding how to manipulate these components through techniques like distribution and combining like terms is essential.
Algebraic expressions form the building blocks of algebra and are used to model real-world problems and relationships.
Polynomial Simplification
Polynomial simplification involves reducing a polynomial to its simplest form. A polynomial is an algebraic expression that consists of variables and coefficients structured as a sum of terms. By simplifying polynomials, you make them more manageable and easier to work with.
In our problem, the polynomial we start with is:
\[4a - a(a + 5)\]
After applying the distributive property and combining like terms, the polynomial simplifies to:
\[-a^2 - a\]
This shows that the expression is reduced to just two terms. Simplification often involves finding and combining like terms, as well as manipulating expressions through rules of arithmetic and algebra to achieve a more straightforward form.

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Most popular questions from this chapter

In the expression below, specify the number of terms that are present, then list them. $$ 3 a(a+1)-(a+2)(a-3) $$

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