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Chapter 5: Problem 54

Multiply vertically. \((4 x-7)(5 x+1)\)

Short Answer

Expert verified
The product is \(20x^2 - 31x - 7\).

Step by step solution

01

Distribute the First Term

We start by distributing the first term from the first binomial, which is \(4x\), across the second binomial. This gives us: - \(4x \times 5x = 20x^2\)- \(4x \times 1 = 4x\)
02

Distribute the Second Term

Next, distribute the second term from the first binomial, which is \(-7\), across the second binomial. This gives us: - \(-7 \times 5x = -35x\)- \(-7 \times 1 = -7\)
03

Combine Like Terms

Now, combine all the terms obtained from distribution: - \(20x^2 + 4x - 35x - 7\).Combine like terms:- The \(x\) terms: \(4x - 35x = -31x\).Final expression: \(20x^2 - 31x - 7\).

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Key Concepts

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

Understanding Binomial Expressions
A binomial expression is a type of polynomial that consists of exactly two terms. These terms are usually separated by a plus or minus sign. In our exercise, the binomials are \((4x - 7)\) and \((5x + 1)\). Each term in the binomial may contain variables, constants, or both.

Some key aspects of binomial expressions include:
  • They have two distinct parts, which makes them easy to identify.
  • The terms in a binomial can be combined or multiplied with other polynomial expressions.
  • One common operation involving binomials is multiplication, which uses the distributive property.
Understanding binomials and how they interact is vital for solving more complex polynomial equations.
Exploring the Distributive Property
The distributive property is a fundamental algebraic concept used to simplify the process of multiplying expressions. It states that if you have three numbers, \(a\), \(b\), and \(c\), then: \(a(b + c) = ab + ac\). This principle helps in distributing the terms of a binomial over another polynomial expression.

In our exercise, we apply the distributive property twice:
  • First, we distribute \(4x\) to both terms of the second binomial \((5x + 1)\).
  • Then, we distribute \(-7\) to both terms of the second binomial as well.
Using this property helps us to ensure that each part of one polynomial is multiplied by every part of the other, facilitating accurate polynomial multiplication.
Combining Like Terms
Combining like terms involves simplifying an expression by gathering all terms that have the same variables raised to the same power. This simplification step is crucial after applying the distributive property, as it reduces the polynomial expression to its simplest form.

In the given exercise:
  • After distributing, we have the terms: \(20x^2\), \(4x\), \(-35x\), and \(-7\).
  • We combine the like terms \(4x\) and \(-35x\) because they both have the variable \(x\).
The final expression becomes \(20x^2 - 31x - 7\). By combining like terms, we make the solution easier to interpret and more efficient for further use in calculations.

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Subtract \(\left(4 y^{2}-6 y-3\right)\) from the sum of \(\left(8 y^{2}+7\right)\) and \((6 y+9)\)

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