/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Find each product. $$ (3 t-4... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 67

Find each product. $$ (3 t-4 s)(t+3 s) $$

Short Answer

Expert verified
The product is \3t^2 + 5ts - 12s^2\.

Step by step solution

01

Distribute the First Term

Multiply the first term of the first binomial, which is \(3t\), with each term of the second binomial. This gives us \(3t \times t = 3t^2\) and \(3t \times 3s = 9ts\).
02

Distribute the Second Term

Multiply the second term of the first binomial, which is \(-4s\), with each term of the second binomial. This gives us \(-4s \times t = -4st\) and \(-4s \times 3s = -12s^2\).
03

Combine Like Terms

Combine the like terms obtained from the previous steps: \(3t^2\), \(9ts\), \(-4st\), and \(-12s^2\). Since \(9ts\) and \(-4st\) are like terms, they can be combined to give \(5ts\). Thus, the final expression is \(3t^2 + 5ts - 12s^2\).

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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 key principle in algebra. It is used to simplify expressions and solve equations. The property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, in the expression \(a(b + c)\), the distributive property allows us to rewrite it as \(ab + ac\). This property is crucial when multiplying binomials, as it helps to distribute each term in the first binomial across every term in the second binomial.
binomials
A binomial is a polynomial with exactly two terms. These terms are separated by a plus (+) or minus (-) sign. Examples of binomials include \(3t - 4s\) and \(t + 3s\). When we multiply binomials, we use various properties, including the distributive property, to expand the expression. Multiplying binomials involves taking each term in the first binomial and distributing it across each term in the second binomial, as shown in the given example.
like terms
Like terms are terms that contain the same variables raised to the same power. For instance, in the expression \(3t^2 + 5ts - 12s^2\), \(5ts\) and \(-4st\) are like terms because they both contain the variables t and s. However, \(3t^2\) and \(-12s^2\) are not like terms because they contain different variables or different powers of the variables. Combining like terms is essential for simplifying algebraic expressions and finding the final solution.

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Most popular questions from this chapter

Before Congress raised the debt limit in \(2017,\) it was \(\$ 1.98 \times 10^{13} .\) To the nearest dollar, how much was this for every man, woman, and child in the country? Use 326 million as the population of the United States. (Data from www.washingtonpost.com; www.census.gov)

Add or subtract as indicated. \((6 b+3 c)+(-2 b-8 c)\)

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 75\. \(\frac{0.00000072(0.00023)}{0.000000018}\)

In 2017 , the population of the United States was about 326.4 million. To the nearest dollar, calculate how much each person in the United States would have had to contribute in order to make one person a trillionaire (that is, to give that person \(\$ 1,000,000,000,000\) ). (Data from U.S. Census Bureau.)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (2 p-5)^{2} $$
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