Problem 72

Question

Use the distributive property to expand $4 x(5 y+11)$.

Step-by-Step Solution

Verified

Answer

Question: Expand the expression $4x(5y+11)$ using the distributive property. Answer: $20xy + 44x$

1Step 1: Apply the distributive property

We need to multiply $4x$ by both terms inside the parentheses. So, we have: $4x(5y+11) = 4x \cdot 5y + 4x \cdot 11$

2Step 2: Multiply the terms

Now, we will multiply the terms as follows: $4x \cdot 5y + 4x \cdot 11 = 20xy + 44x$

3Step 3: Write the final result

The expanded expression using the distributive property is: $4x(5y+11) = 20xy + 44x$

Key Concepts

Algebraic ExpressionsMultiplicationExpansionPolynomials

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and mathematical operators like addition and multiplication. In our exercise, the expression $4x(5y+11)$ involves variables $x$ and $y$, numbers 4, 5, and 11, and operations of multiplication and addition. These expressions are essential because they allow us to represent relationships and changes mathematically.

Numbers: Constants that do not change (e.g., 4, 5, 11).
Variables: Symbols that represent unknown numbers (e.g., $x, y$).
Terms: Parts of the expression separated by addition or subtraction (e.g., $4x \, ext{and} \, 5y$).

Understanding algebraic expressions is crucial as it forms the foundation for solving equations and performing various algebraic operations.

Multiplication

Multiplication is one of the basic arithmetic operations used to find the product of two numbers or expressions. It is a shortcut for repeated addition. In algebra, multiplication is often used to simplify and expand expressions.In our problem, $4x$ is multiplied by each term within the parentheses $(5y + 11)$. This means:

Multiply $4x$ by $5y$, resulting in $20xy$.
Multiply $4x$ by 11, resulting in $44x$.

Multiplication should be performed with care, especially when dealing with variables, as it requires combining both coefficients and variables correctly.

Expansion

Expansion in algebra refers to the process of removing parentheses by multiplying the terms inside the parentheses with those outside. This process, also known as using the distributive property, allows us to express an algebraic expression in a different form.To expand $4x(5y+11)$ using the distributive property, we follow these steps:

Distribute $4x$ to both $5y$ and 11.
Compute $4x \cdot 5y = 20xy$.
Compute $4x \cdot 11 = 44x$.

Through expansion, complex algebraic expressions become simpler to work with, enabling further manipulation and simplification.

Polynomials

Polynomials are a type of algebraic expression that incorporates variables raised to whole number exponents and their coefficients. They are typically summed up terms like $ax^n + bx^{n-1} + \ldots + c$. Understanding polynomials is important for many areas within mathematics and applied sciences.The expanded expression from the exercise, $20xy + 44x$, is an example of a polynomial:

Polynomial Terms: $20xy$ and $44x$
Variables: $x, y$
Coefficients: 20 for $xy$ term and 44 for $x$ term

By learning how to handle polynomials, we gain the ability to solve complex mathematical problems involving equations, functions, and more.

Problem 72

Other exercises in this chapter

Problem 72

Convert the following problems from scientific form to standard form. $$ 4.116 \times 10^{-2} $$

View solution

Problem 72

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ -4 a^{3} b^{-5}\left(2 a^{2} b^{7} c^{-2}\right) $$

View solution

Problem 72

Find the sums for the the following problems. $[10+(-6)]+[12+(-2)]$

View solution

Problem 73

Perform the following operations. $$ \left(8.8 \times 10^{-50}\right)^{2} $$

View solution