Problem 48 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (9 x+1)(-7) $$

Step-by-Step Solution

Answer

The expression (9x + 1)(-7) without parentheses is -63x - 7.

Step 1: Identify the values of a, x and b

Here, a = -7, x = 9x and b = 1.

Step 2: Apply the distributive property

Applying the distributive property ax + ab gives -7*9x + -7*1.

Step 3: Simplify the resulting terms

This simplifies to -63x -7.

Key Concepts

Simplifying Expressions

To understand the process of simplifying expressions, consider it as a method of making algebraic expressions more manageable and easier to work with. The goal is to combine like terms and use the appropriate algebraic rules to simplify the expression to its most basic form without changing its value. In the exercise \( (9x + 1)(-7) \), we are asked to simplify an expression involving parentheses by distributing a single term over a binomial.

When faced with an expression that includes a binomial – two terms connected by a plus or minus sign – the distributive property allows us to remove the parentheses by multiplying each term inside the parentheses by the term outside. Here, \( -7 \) is distributed across \( 9x \) and \( 1 \) leading to two simpler expressions: \( -7 \cdot 9x \) and \( -7 \cdot 1 \). This step is crucial as it breaks down a complex expression into simpler components that we can easily combine or further simplify.

After performing the multiplication, we combine the resulted expressions to obtain the simplified form of the original expression. Hence, mastering the ability to simplify algebraic expressions is fundamental for solving more complex equations and understanding algebra as a whole.

Algebraic Properties

Algebraic properties serve as the ruleset for manipulating and solving algebraic expressions. One of the foundational properties is the distributive property, which is showcased in the exercise mentioned above. This property states that for any real numbers \( a \) , \( b \) , and \( c \), the equation \( a(b + c) = ab + ac \) holds true.

The demonstration of this property through a practical example helps reinforce understanding. By considering \( (9x + 1)(-7) \), and distributing \( -7 \) to both \( 9x \) and \( +1 \) individually, we apply the distributive property directly. It’s important to manage the sign according to multiplication rules: a negative number multiplied by a positive number results in a negative number.

Moreover, through multiple examples and applications of the distributive property, one can gain a clearer insight into how algebraic expressions are manipulated. This understanding is pivotal since it forms the basis upon which more complex algebraic operations are constructed, such as working with exponents and solving equations.

Multiplying Polynomials

Multiplying polynomials can seem daunting at first, but by using the distributive property and combining like terms, it becomes a systematic process. To multiply polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial. If we consider the given exercise, while it doesn’t involve two polynomials, it provides a stepping stone towards understanding polynomial multiplication.

In our case, \( (9x + 1) \) acts as the first 'polynomial', and \( -7 \) is the single term we're distributing across it. The process mirrors that of multiplying two binomials, only simplified. Once each term has been multiplied, the next step is to combine like terms, if any, to reach the simplest form of the resulting polynomial. In the given exercise, no combining is necessary since \( -63x \) and \( -7 \) are not like terms.

Understanding how to multiply single terms by each term in a polynomial is an integral skill in algebra. It leads to mastery in multiplying larger polynomials together, where the distributive property is applied multiple times. This concept is also a precursor to more advanced topics in algebra, such as factoring polynomials and finding polynomial roots.