/*! 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 23 Expand each expression. $$ 3... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 23

Expand each expression. $$ 3(3 a-7) $$

Short Answer

Expert verified
The expanded form is \( 9a - 21 \).

Step by step solution

01

Identify the Expression

The given expression is \( 3(3a - 7) \). This is a single-term factor (3) multiplied by a binomial (\(3a - 7\)).
02

Apply the Distributive Property

Use the distributive property, which states that \( a(b + c) = ab + ac \), to multiply the factor outside the parentheses by each term inside the parentheses.
03

Distribute the Factor

Multiply 3 by each term inside the parentheses: \[ 3 \times (3a) - 3 \times (7) \].
04

Simplify Each Term

Perform the multiplication for each term: \[ 3 \times 3a = 9a \] and \[ 3 \times (-7) = -21 \].
05

Combine the Results

Combine the simplified terms to write the expanded expression as \[ 9a - 21 \].

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Key Concepts

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

expanding expressions
Many algebraic problems require expanding expressions. This means taking an expression in a compact form and breaking it down into a more extended form by distributing and simplifying it. For example, with the expression \(3(3a-7)\), we use the distributive property. First, we multiply the single-term factor outside the parentheses by each term inside the parentheses. Expanding expressions helps to simplify and solve equations more easily and understand each component better.
multiplication of binomials
The multiplication of binomials is a common task when expanding expressions. A binomial is an expression that contains two terms, like \(3a - 7\). When you multiply it by a single-term factor like 3, you distribute the 3 to each term within the binomial. This means you perform the operation 3 times \(3a\) and 3 times \(-7\). By doing so, you end up with two separate multiplication products which you then simplify. This method is essential for handling more complex algebraic equations.
algebraic simplification
Algebraic simplification involves making an expression as simple as possible. After expanding expressions, you often face multiple terms that can be simplified. In our example, multiplying \(3(3a - 7)\) results in \(9a - 21\). Here, we multiply first, then simplify the terms into one neat expression. Simplification makes equations easier to solve and understand by breaking down complex problems into simpler parts. It's a crucial step in all algebraic operations.

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Most popular questions from this chapter

Because 7 isn't a perfect square, the expression \(4 x^{2}-7\) doesn't look like the difference of two squares. But 7 is the square of something. a. What is 7 the square of? b. How can you use your answer to Part a to factor \(4 x^{2}-7\) into a product of two binomials?

Physical Science Suppose that, at some point into its flight, a particular rocket's height \(h,\) in meters, above sea level \(t\) seconds after launching depends on \(t\) according to the formula \(h=2 t(60-t)\) a. How many seconds after launching will the rocket return to sea level? b. Write and solve an equation to find when the rocket will be \(1,200 \mathrm{m}\) above sea level.

In Exercises \(5-14\) , determine whether the expression on the left of the equal sign is a difference of squares or a perfect square trinomial. If is, indicate which and then factor the expression and solve the equation for \(x\) . If the expression is in neither form, say so. $$ x^{2}-14 x+49=0 $$

Solve the equation \(9 x^{2}-16=0\) by factoring and by using the quadratic formula.

Factor the expression on the left side of each equation as much as possible, and find all the possible solutions. It will help to remember that \(x^{4}=\left(x^{2}\right)^{2}, x^{8}=\left(x^{4}\right)^{2},\) and \(x^{3}=x\left(x^{2}\right) .\) $$ x^{4}+2 x^{2}+1=0 $$
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