Problem 78 Multiply. $$ (8 x-1)(3 x-7) ... [FREE SOLUTION]

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Chapter 5: Problem 78

Multiply. $$ (8 x-1)(3 x-7) $$

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Introduction to Distribution

To multiply the expressions $(8x - 1)(3x - 7)$, use the distributive property \

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term inside a set of parentheses. In simple terms, it means distributing the multiplication over addition or subtraction inside the parentheses. For example, if you have an expression like $(a(b + c))$, you can distribute to get $(ab + ac)$. This approach makes it easier to handle multiple terms in an expression and ensures everything is covered without missing any critical parts.

In the case of the problem $(8x - 1)(3x - 7)$, you would distribute as follows:

First, multiply $8x$ by both $3x$ and $-7$ in the second set of parentheses.
Then, multiply $-1$ by both $3x$ and $-7$.

This gives us four separate multiplications, which we can then combine to simplify the expression.

Polynomial Expansion

Polynomial expansion refers to the process of multiplying together polynomials, which are algebraic expressions consisting of variables and coefficients. Expanding polynomials often involves using the distributive property across multiple terms.

When you have two binomials, like our example $(8x - 1)(3x - 7)$, expanding them involves:

First, expanding by multiplying every term from the first binomial with every term from the second.
Then, simplifying the resulting expression.

In our case, you handle the following calculations:

Multiply $8x$ with $3x$ to get $24x^2$.
Multiply $8x$ with $-7$ to get $-56x$.
Multiply $-1$ with $3x$ to get $-3x$.
Multiply $-1$ with $-7$ to get $7$.

Combining all these gives the expanded polynomial $24x^2 - 56x - 3x + 7$, which simplifies further by combining like terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They are used to represent real-life problems or mathematical relationships in a simplified form. When dealing with expressions, variables stand for unknown or changeable values, while coefficients are constants that multiply the variables.

In our problem, $(8x - 1)(3x - 7)$, each bracket contains an algebraic expression.

The first expression is $8x - 1$, where $8x$ is the term with the variable $x$, and $-1$ is the constant.
The second expression $3x - 7$ has a similar structure, with $3x$ as the variable term and $-7$ as the constant.

When multiplying such expressions, it is critical to keep track of each term and its sign. Clearingly understanding and manipulating algebraic expressions is essential to algebra, as it helps solve various mathematical problems systematically.

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91影视

Multiply. $$ (8 x-1)(3 x-7) $$

Short Answer

Step by step solution

Introduction to Distribution

Key Concepts

Distributive Property

Polynomial Expansion

Algebraic Expressions

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