Problem 56 Find each product. $$(3 a-2 b)... [FREE SOLUTION]

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Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 56

Find each product. $$(3 a-2 b)(4 a+b)$$

Short Answer

Expert verified

The result of the product $(3 a-2 b)(4 a+b)$ is $12a^2 - 5ab - 2b^2$

Step by step solution

Distribute the first term of the first binomial

Multiply the first term of the first binomial $3a$ by each term in the second binomial. This gives: $3a * 4a = 12a^2$ and $3a * b = 3ab$

Distribute the second term of the first binomial

Multiply the second term of the first binomial $-2b$ by each term in the second binomial. This gives: $-2b * 4a = -8ab$ and $-2b * b = -2b^2$

Simplify

Combine like terms and write the result in the standard form of a quadratic equation $ax^2+bx+c$. This gives: $12a^2 + 3ab - 8ab - 2b^2 = 12a^2 - 5ab - 2b^2$

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Key Concepts

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

Understanding Binomials

Binomials are expressions that contain exactly two terms separated by a plus (+) or minus (-) sign. They play an important role in many algebraic operations, including multiplication. An example of a binomial is $3a - 2b$, which has two distinct terms: $3a$ and $-2b$. Understanding how to manipulate these expressions is key to solving equations and finding products.

When handling binomials, it's crucial to recognize each term as a separate entity with its own coefficient and variable. For instance, in the term $3a$, $3$ is the coefficient and $a$ is the variable. This simple understanding allows us to apply operations like multiplication with clarity and precision.

Binomials consist of two terms.
They can be added, subtracted, or multiplied.
Each term can have coefficients and variables.

What is a Quadratic Equation?

A quadratic equation is a type of polynomial equation that involves variables raised to the second power. The standard form of a quadratic equation is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

When multiplying binomials, you often end up with a quadratic expression. This is because the highest degree in the resulting expression is two, stemming from terms like $a^2$ when two binomials are multiplied together. It's this feature that categorizes the outcome as a quadratic equation.

Quadratic equations contain terms with a variable squared ($x^2$).
They often result from multiplying binomials.
The standard format is $ax^2 + bx + c$.

The Distributive Property Explained

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term within a set of parentheses. It's useful when multiplying binomials, as it ensures each term is multiplied appropriately.

For example, in the expression $(3a - 2b)(4a + b)$, the distributive property is used to multiply each term in the first binomial with each term in the second binomial.

First, distribute $3a$ across $4a + b$, resulting in $12a^2 + 3ab$.
Next, distribute $-2b$ across $4a + b$, resulting in $-8ab - 2b^2$.
Finally, combine like terms to simplify the expression.

By understanding and applying the distributive property, you can confidently solve more complex algebraic equations involving binomials.

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91影视

Find each product. $$(3 a-2 b)(4 a+b)$$

Short Answer

Step by step solution

Distribute the first term of the first binomial

Distribute the second term of the first binomial

Simplify

Key Concepts

Understanding Binomials

What is a Quadratic Equation?

The Distributive Property Explained

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