Problem 93 Write an equivalent expression b... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 93

Write an equivalent expression by factoring. $$3 a^{n+1}+6 a^{n}-15 a^{n+2}$$

Short Answer

Expert verified

3a^n (2 - 4a)

Step by step solution

- Identify the common factor

Examine each term in the given expression: - First term: 3a^{n+1} - Second term: 6a^n - Third term: -15a^{n+2} Notice that each term has a factor of 3 and a factor of a^n.

- Factor out the greatest common factor (GCF)

Extract the greatest common factor, which is 3a^n, from each term:\[ 3a^n( a + 2 - 5a ) \]

- Simplify inside the parentheses

Simplify the expression inside the parentheses:\[ 3a^n (a + 2 - 5a) \]Combine like terms:\[ 3a^n (-4a + 2) \]Which simplifies to:\[ 3a^n (2 - 4a) \]

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Key Concepts

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

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that can evenly divide each term in an expression. In our exercise, we started with the expression: \[3 a^{n+1}+6 a^n-15 a^{n+2}\]First, examine each term individually. Let's break it down:

First term: $3a^{n+1}$
Second term: $6a^n$
Third term: $-15a^{n+2}$

When looking for the GCF among these terms, notice that each term has both a numerical factor and a common variable factor. Each term includes a numerical factor of 3, and since the smallest exponent is $a^n$, each term also has a common variable factor of $a^n$. Therefore, the GCF in this case is $3a^n$. By factoring $3a^n$ out of each term, we simplify the expression and make it easier to work with.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form by combining like terms or factoring. After identifying the GCF in our example expression, we factor it out:\[3a^n(a + 2 - 5a)\]Now, focus on simplifying the expression inside the parentheses. Our goal is to combine like terms. By analyzing the terms inside the parentheses:

$a$
$2$
$-5a$

We can see that $a$ and $-5a$ are like terms. Combine them to simplify:\[3a^n(a - 5a + 2) = 3a^n(-4a + 2)\]Thus, the simplified expression is:\[3a^n(2 - 4a)\]Through this process, we transformed our original polynomial into a simpler, equivalent expression.

Term Identification

Term identification is essential in the simplification process. Each term in a polynomial is separated by addition or subtraction. In our given polynomial, terms include:

$3a^{n+1}$
$6a^n$
$-15a^{n+2}$

Identifying the terms correctly allows us to factor out the GCF and combine like terms efficiently. In this exercise:

$3a^{n+1}$
$6a^n$
$-15a^{n+2}$

Helps us recognize the common variable factor $a^n$, and the consistent numerical factor 3. By isolating and factoring these out, we make the expression more manageable. Correctly identifying terms is the foundation of any polynomial simplification process.

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91影视

Write an equivalent expression by factoring. $$3 a^{n+1}+6 a^{n}-15 a^{n+2}$$

Short Answer

Step by step solution

- Identify the common factor

- Factor out the greatest common factor (GCF)

- Simplify inside the parentheses

Key Concepts

Greatest Common Factor (GCF)

Polynomial Simplification

Term Identification

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