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Chapter 10: Problem 9

Find the product. $$ (4 x+7)(-2 x) $$

Short Answer

Expert verified
The product of (4x + 7) and (-2x) is -8x^2 - 14x.

Step by step solution

01

Distribute the monomial to each term in the binomial

Multiply \(-2x\) by both items in the parenthesis \((4x+7)\). This gives \( -2x*4x + -2x*7\).
02

Simplify

The product of \(-2x\) and \(4x\) is \(-8x^2\), and the product of \(-2x\) and \(7\) is \(-14x\). So, in total, we have \(-8x^2 - 14x\).

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Key Concepts

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

Understanding Monomials
A monomial is a single algebraic term that can consist of a number, variable, or a combination of both. In mathematical expression terms, monomials are the simplest form of polynomials. For example, in the expression
  • \(-2x\) is a monomial.
  • It consists of a coefficient, which is \(-2\), and a variable, \(x\).
  • The degree of a monomial is determined by the total number of times the variable is multiplied by itself, which in this case is 1.
Monomials can be multiplied by other monomials, binomials, or polynomials using the distributive property. This is because monomials are straightforward, allowing operations like multiplication to spread over terms inside parenthesis. Knowing how to handle monomials is foundational for simplifying and tackling more complex expressions.
Exploring Binomials
A binomial is an algebraic expression that contains two terms, which are typically summed or subtracted. Understanding binomials involves recognizing their structure and how they interact with other algebraic expressions. For example:
  • In \((4x + 7)\), we have a binomial.
  • There are two distinct terms: \(4x\) and \(7\).
  • Each term can have its own coefficient and variable, affecting how the binomial reacts during multiplication.
When a monomial multiplies a binomial, the distributive property is used. This means the monomial will multiply with each term in the binomial separately, which is a crucial concept in simplifying products of algebraic expressions. Navigating binomials efficiently requires practice in recognizing their structure and applying operations properly.
Introduction to Polynomials
Polynomials are algebraic expressions that consist of one or more terms. They include monomials, binomials, and trinomials, reflecting the number of terms involved. When working with polynomials, it's essential to be able to perform operations like addition, subtraction, and especially multiplication. Here are the basics:
  • A polynomial can have any number of terms, so long as they involve whole numbers as the variable's exponent.
  • They can be simple, like a monomial, or complex with multiple terms, like a trinomial.
  • Understanding distribution is key in polynomial multiplication. This means every term must be multiplied appropriately.
In solving algebraic problems, polynomials frequently require simplification by multiplication. The understanding of the distributive property aids in tackling polynomial expressions, especially when multiplying by a monomial or another polynomial. Mastery of these basics makes handling more complex algebraic expressions much more approachable.

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