Problem 61
Question
For the following problems, use the distributive property to expand the quantities. $$15 x(2 y+3 z)$$
Step-by-Step Solution
Verified Answer
Answer: The expanded form of the expression is \(30xy + 45xz\).
1Step 1: Identify the factors in the given expression
In this expression, the factors are \(15x\) and \((2y+3z)\). We need to apply the distributive property to these factors.
2Step 2: Apply the distributive property
According to the distributive property, we will multiply \(15x\) with both terms inside the parentheses, \((2y+3z)\). This can be written as:
$$15x * (2y + 3z) = (15x * 2y) + (15x * 3z)$$
3Step 3: Multiply the terms
Now, we will multiply the terms in the expression:
$$(15x * 2y) + (15x * 3z) = 30xy + 45xz$$
The expression is now expanded using the distributive property, and the final result is:
$$15x(2y + 3z) = 30xy + 45xz$$
Key Concepts
Algebraic ExpressionMultiplying PolynomialsCombining Like Terms
Algebraic Expression
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Variables are symbols, often letters like x, y, and z, that represent numbers that can change (unknowns).
For instance, in the expression \(15x(2y + 3z)\), 15x is a coefficient (a constant multiplier), and the variables are x, y, and z. Operation symbols in this case include multiplication (indicated by juxtaposition) and addition (the plus sign). Algebraic expressions are important in forming equations and modeling real-world situations.
Understanding how to manipulate these expressions using various properties is a crucial skill in algebra. When dealing with algebraic expressions, students often perform operations such as simplification, factorization, and expansion, each of which plays a significant role in solving complex algebraic problems.
For instance, in the expression \(15x(2y + 3z)\), 15x is a coefficient (a constant multiplier), and the variables are x, y, and z. Operation symbols in this case include multiplication (indicated by juxtaposition) and addition (the plus sign). Algebraic expressions are important in forming equations and modeling real-world situations.
Understanding how to manipulate these expressions using various properties is a crucial skill in algebra. When dealing with algebraic expressions, students often perform operations such as simplification, factorization, and expansion, each of which plays a significant role in solving complex algebraic problems.
Multiplying Polynomials
Multiplying polynomials involves combining two or more algebraic expressions using the distributive property. The distributive property states that for any three numbers (or expressions), a, b, and c, \(a(b + c) = ab + ac\).
For example, to multiply the polynomials \(15x\) and \((2y + 3z)\), you distribute the \(15x\) across the terms within the parentheses:
\[15x(2y + 3z) = (15x \times 2y) + (15x \times 3z)\]
This process eventually leads to an expansion of the product into a larger expression where like terms can be combined if they exist. Since polynomials can range from simple binomials to more complex expressions with several terms, the distributive property is a fundamental concept in ensuring that every term is properly accounted for during multiplication.
For example, to multiply the polynomials \(15x\) and \((2y + 3z)\), you distribute the \(15x\) across the terms within the parentheses:
\[15x(2y + 3z) = (15x \times 2y) + (15x \times 3z)\]
This process eventually leads to an expansion of the product into a larger expression where like terms can be combined if they exist. Since polynomials can range from simple binomials to more complex expressions with several terms, the distributive property is a fundamental concept in ensuring that every term is properly accounted for during multiplication.
Combining Like Terms
Combining like terms is a process used to simplify algebraic expressions or equations. 'Like terms' refer to terms that have the exact same variable parts, meaning the variables and their exponents are identical.
In the expression \(30xy + 45xz\), there are no like terms because every term contains a unique set of variables. However, if we had an expression like \(30xy + 15xy\), we could combine the like terms to get \(45xy\) as both terms include the variable part xy with the same exponents.
When combining like terms, we simply add or subtract the coefficients, which are the numerical parts of the terms, while keeping the variable part constant. This process is crucial after using the distributive property, as it leads to a more simplified and concise expression, which is easier to work with for subsequent algebraic operations.
In the expression \(30xy + 45xz\), there are no like terms because every term contains a unique set of variables. However, if we had an expression like \(30xy + 15xy\), we could combine the like terms to get \(45xy\) as both terms include the variable part xy with the same exponents.
When combining like terms, we simply add or subtract the coefficients, which are the numerical parts of the terms, while keeping the variable part constant. This process is crucial after using the distributive property, as it leads to a more simplified and concise expression, which is easier to work with for subsequent algebraic operations.
Other exercises in this chapter
Problem 61
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