Problem 6
Question
a. Find the GCF of \(30 x^{2}\) and \(105 x^{3}\). $$ \begin{array}{l} {30 x^{2}=2 \cdot 3 \cdot 5 \cdot x \cdot x} \\ {105 x^{3}=3 \cdot 5 \cdot 7 \cdot x \cdot x \cdot x} \end{array} $$ b. Find the GCF of \(12 a^{2} b^{2}, 15 a^{3} b,\) and \(75 a^{4} b^{2}\) $$ \begin{array}{l} {12 a^{2} b^{2}=2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot b \cdot b} \\ {15 a^{3} b=3 \cdot 5 \cdot a \cdot a \cdot a \cdot b} \\ {75 a^{4} b^{2}=3 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b} \end{array} $$
Step-by-Step Solution
Verified Answer
a. GCF is \(15x^2\). b. GCF is \(3a^2b\).
1Step 1: Prime Factorization for Part a
First, let's perform the prime factorization for each term. For \(30x^2\), we have the factorization: \[30 x^{2} = 2 \cdot 3 \cdot 5 \cdot x \cdot x.\] For \(105x^3\), the factorization is \[105 x^{3} = 3 \cdot 5 \cdot 7 \cdot x \cdot x \cdot x.\]
2Step 2: Identify Common Factors for Part a
Look at the prime factors of each term, and list common factors. For \(30x^2\) and \(105x^3\), the common factors are: \[3,\ 5,\ x,\ x.\]
3Step 3: Calculate the GCF for Part a
Multiply the common prime factors to find the GCF. \[GCF = 3 \cdot 5 \cdot x \cdot x = 15x^2.\]
4Step 4: Prime Factorization for Part b
Perform prime factorization for each expression. For \(12a^2b^2\), \[12 a^{2} b^{2} = 2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot b \cdot b.\] For \(15a^3b\),\[15 a^{3} b = 3 \cdot 5 \cdot a \cdot a \cdot a \cdot b.\] For \(75a^4b^2\),\[75 a^{4} b^{2} = 3 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b.\]
5Step 5: Identify Common Factors for Part b
List out the common factors among the three terms. The common prime factors among the three are: \[3,\ a,\ a,\ b.\]
6Step 6: Calculate the GCF for Part b
Multiply these common prime factors to obtain the GCF. \[GCF = 3 \cdot a \cdot a \cdot b = 3a^2b.\]
Key Concepts
Prime FactorizationAlgebraic ExpressionsCommon Factors
Prime Factorization
Prime factorization is the process of breaking down a number into its smallest prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This method is very useful in finding the greatest common factor (GCF) of two or more algebraic expressions.
To perform prime factorization, divide the number by the smallest possible prime number and continue this process with the quotient, until all factors are prime numbers.
For example, for the number 30, the prime factorization would be 2, 3, and 5. Similarly, in algebraic expressions, both numerical coefficients and variables are considered. For instance, the term \(30x^2\) breaks down to factors of 2, 3, 5, and \(x \times x\).
Once prime factors are identified, finding common elements from different expressions becomes fairly straightforward. This method simplifies complexity in calculations and ensures accurate outcomes.
To perform prime factorization, divide the number by the smallest possible prime number and continue this process with the quotient, until all factors are prime numbers.
For example, for the number 30, the prime factorization would be 2, 3, and 5. Similarly, in algebraic expressions, both numerical coefficients and variables are considered. For instance, the term \(30x^2\) breaks down to factors of 2, 3, 5, and \(x \times x\).
Once prime factors are identified, finding common elements from different expressions becomes fairly straightforward. This method simplifies complexity in calculations and ensures accurate outcomes.
Algebraic Expressions
Algebraic expressions are mathematical phrases that contain numbers, variables, and operation symbols. They are the backbone of algebra and allow us to represent real-world situations abstractly using letters for numbers.
An expression like \(105x^3\) is made up of a coefficient (105) and a variable raised to a power (\(x^3\)). In algebraic expressions, different rules apply when performing operations like addition, multiplication, or factorization.
Recognizing individual elements within an expression, such as coefficients and variables, helps to simplify and solve mathematical problems. Algebraic expressions can vary greatly in complexity, from monomials like \(15a^3b\) to more complicated polynomials. The beauty of these expressions lies in their ability to be manipulated through various algebraic operations to find solutions to equations or inequalities.
Understanding each component within an algebraic expression makes tasks like evaluating expressions, simplifying, and solving equations much more manageable.
An expression like \(105x^3\) is made up of a coefficient (105) and a variable raised to a power (\(x^3\)). In algebraic expressions, different rules apply when performing operations like addition, multiplication, or factorization.
Recognizing individual elements within an expression, such as coefficients and variables, helps to simplify and solve mathematical problems. Algebraic expressions can vary greatly in complexity, from monomials like \(15a^3b\) to more complicated polynomials. The beauty of these expressions lies in their ability to be manipulated through various algebraic operations to find solutions to equations or inequalities.
Understanding each component within an algebraic expression makes tasks like evaluating expressions, simplifying, and solving equations much more manageable.
Common Factors
Common factors are the shared factors between two or more numbers or algebraic expressions. They hold essential significance when aiming to simplify expressions or find the greatest common factor, which simplifies the problem-solving process.
To find the common factors, first perform the prime factorization of each number or expression. Identifying which prime factors appear in each group will lead to finding the common ones. Let's take the solution steps we have; the common factors of the expressions \(30x^2\) and \(105x^3\) are 3, 5, and \(x \times x\). These repeated elements are the building blocks for calculating the GCF.
For multiple expressions as seen in the second part, consistently be on the lookout for overlapping factors across all terms. In our example, \(12a^2b^2\), \(15a^3b\), and \(75a^4b^2\) share the factors 3, \(a \times a\), and \(b\). Without common factors, finding a GCF would not be feasible, which in turn complicates simplifying or solving algebraic problems.
To find the common factors, first perform the prime factorization of each number or expression. Identifying which prime factors appear in each group will lead to finding the common ones. Let's take the solution steps we have; the common factors of the expressions \(30x^2\) and \(105x^3\) are 3, 5, and \(x \times x\). These repeated elements are the building blocks for calculating the GCF.
For multiple expressions as seen in the second part, consistently be on the lookout for overlapping factors across all terms. In our example, \(12a^2b^2\), \(15a^3b\), and \(75a^4b^2\) share the factors 3, \(a \times a\), and \(b\). Without common factors, finding a GCF would not be feasible, which in turn complicates simplifying or solving algebraic problems.
Other exercises in this chapter
Problem 6
Fill in the blanks. \(x^{2}+x-56=(x \square)(x \square\))
View solution Problem 6
Fill in the blanks. a. \(x^{3}+64 y^{3}=(\quad)^{3}+(\quad)^{3}\) b. \(8 x^{3}-27=(\quad)^{3}-(\quad)^{3}\)
View solution Problem 7
For each of the following polynomials, which factoring method would you use first? $$ x^{3}+27 $$
View solution Problem 7
Multiply both sides of the equation by \(2 .\) Do not solve. $$ 10=\frac{1}{2} b(b+5) $$
View solution