Problem 45
Question
For the following problems, list, if any should appear, the common factors in the expressions. $$ 9 a(a-3)^{2}+10 b(a-3) $$
Step-by-Step Solution
Verified Answer
Answer: The common factor in the given expression is \((a-3)\).
1Step 1: Write down the given expression
We have the expression:
$$
9 a(a-3)^{2}+10 b(a-3)
$$
2Step 2: Identify the terms in each part of the sum
The expression can be rewritten as the sum of two terms:
$$
\text{Term 1: } 9a(a-3)^2
$$
$$
\text{Term 2: } 10b(a-3)
$$
3Step 3: Factor out the common expression
As we can see in both terms, the common expression is \((a-3)\). So we can factor it out of the sum:
$$
(a-3)(9a(a-3) + 10b)
$$
4Step 4: Simplify the expression if possible
In this case, there are no additional common factors or expressions in the remaining terms. Hence, the final expression is:
$$
(a-3)(9a(a-3) + 10b)
$$
In conclusion, the common factor in the given expression is \((a-3)\).
Key Concepts
Common FactorsAlgebraic TermsSimplifying Expressions
Common Factors
In algebra, identifying common factors is akin to finding threads that connect various parts of a fabric. These factors act as a bridge, tying together different algebraic terms within an expression. For instance, when we examine the expression \[9a(a-3)^2 + 10b(a-3)\], we can unravel this algebraic tapestry by spotting the shared factor that weaves through the terms; in this case, the factor \((a-3)\).
Upon identifying it, we realize that this commonality exists in each term of the sum; this is akin to a repetitive pattern in our fabric metaphor. By carefully extracting this factor, we can significantly simplify the expression. This process not only makes the expression cleaner and more concise but also prepares it for further algebraic manipulation, such as solving for unknown variables or evaluating its value. Remember that the greatest common factor, much like the strongest thread, holds the utmost importance, as it is the largest expression that divides each term without any remainder.
Upon identifying it, we realize that this commonality exists in each term of the sum; this is akin to a repetitive pattern in our fabric metaphor. By carefully extracting this factor, we can significantly simplify the expression. This process not only makes the expression cleaner and more concise but also prepares it for further algebraic manipulation, such as solving for unknown variables or evaluating its value. Remember that the greatest common factor, much like the strongest thread, holds the utmost importance, as it is the largest expression that divides each term without any remainder.
Algebraic Terms
Algebraic terms are the individual building blocks that make up an algebraic expression, much like how bricks are to a building. Each term is a combination of numbers, variables (like our friends 'a' and 'b'), and sometimes exponents, as seen in \(9a(a-3)^2\) and \(10b(a-3)\) within the given exercise.
To understand the structure of algebraic expressions, one must look at these terms individually. The first term, \(9a(a-3)^2\), hides a multiplication of a constant (9), a variable (a), and a binomial raised to a power ((a-3)²). The second term, \(10b(a-3)\), simplifies to the multiplication of a constant (10), a different variable (b), and a binomial. These terms have distinct identity but can sometimes share common components that allow us to group them together efficiently—this is where factoring becomes an essential skill in algebra.
To understand the structure of algebraic expressions, one must look at these terms individually. The first term, \(9a(a-3)^2\), hides a multiplication of a constant (9), a variable (a), and a binomial raised to a power ((a-3)²). The second term, \(10b(a-3)\), simplifies to the multiplication of a constant (10), a different variable (b), and a binomial. These terms have distinct identity but can sometimes share common components that allow us to group them together efficiently—this is where factoring becomes an essential skill in algebra.
Simplifying Expressions
When it comes to simplifying expressions, think of it as decluttering a room. Just as you would group similar items together to organize your space, simplifying an algebraic expression involves combining like terms and factoring out common factors to make the expression more manageable.
In our exercise, after factoring out the common factor \((a-3)\), we are left with the simplified form \((a-3)(9a(a-3) + 10b)\). It's important to note that simplification might not always mean making the expression shorter. The goal is to transform it into a form that is easier to work with for subsequent operations. Think of it as an organizer's finishing touch that, while the room may not end up with fewer items, they're arranged in a way that makes them easier to handle. The beauty of simplifying expressions lies in the clarity it brings to the problem at hand, allowing further algebraic procedures to proceed with less obstruction.
In our exercise, after factoring out the common factor \((a-3)\), we are left with the simplified form \((a-3)(9a(a-3) + 10b)\). It's important to note that simplification might not always mean making the expression shorter. The goal is to transform it into a form that is easier to work with for subsequent operations. Think of it as an organizer's finishing touch that, while the room may not end up with fewer items, they're arranged in a way that makes them easier to handle. The beauty of simplifying expressions lies in the clarity it brings to the problem at hand, allowing further algebraic procedures to proceed with less obstruction.
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Problem 45
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