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Chapter 5: Problem 46

\(\left(z^{4}-2 z^{2}\right)+3\left(z^{4}+4\right)\)

Short Answer

Expert verified
The simplified polynomial is \(4z^{4} - 2z^{2} + 12\).

Step by step solution

01

Distribution

Distribute the 3 in the second group to both terms, resulting in \(3z^{4}\) and 12.
02

Combine Like Terms

Merge similar terms together. In particular, merge \(z^{4}\) terms and then constant terms. This gives \(z^{4} + 3z^{4}\) (the \(z^{4}\) terms) and \(-2z^{2} \) and \(+12\) (the constant terms).
03

Simplify

Adding the coefficients for \(z^{4}\), you get \(4z^{4}\). Add the constants, you get \(-2z^{2} + 12\).

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Key Concepts

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

Distribution in Algebra
In algebra, distribution is a key concept that helps us simplify expressions. It involves multiplying each term within a set of parentheses by a number or another term outside the parentheses. This process is often called the distributive property.

In the expression \( \left(z^{4}-2z^{2}\right)+3\left(z^{4}+4\right) \), distribution occurs with the term 3 multiplying each term inside the parentheses \((z^{4}+4)\).

  • Multiply 3 by each term inside the parentheses: \(3 \times z^{4} = 3z^{4}\) and \(3 \times 4 = 12\).
This step transforms the expression into \(z^{4} - 2z^{2} + 3z^{4} + 12\).

Distributing terms helps in breaking down complex expressions into more manageable parts. It鈥檚 a foundational skill in simplifying algebraic expressions.
Combining Like Terms
Combining like terms is another vital technique in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers but can have different coefficients. By combining them, you reduce the complexity of the expression.

In the expression from distribution, \(z^{4} - 2z^{2} + 3z^{4} + 12\), we identify the like terms:

  • \(z^{4}\) and \(3z^{4}\) are like terms because they both contain \(z^{4}\).
  • \(-2z^{2}\) is a separate term as it has a different exponent.
  • The constant term is 12.
To combine like terms, add their coefficients:

  • \(1z^{4} + 3z^{4} = 4z^{4}\)
Now the expression becomes \(4z^{4} - 2z^{2} + 12\).

Combining like terms condenses expressions, making them easier to work with.
Simplifying Expressions
Simplifying expressions is the process of reducing an expression to its simplest form. This makes it easier to understand and work with for solving equations or inequalities.

After distributing and combining like terms, you are already on your way to simplification. Our simplified expression is now \(4z^{4} - 2z^{2} + 12\).

Here鈥檚 a quick check to ensure it's fully simplified:

  • Ensure that all like terms are combined.
  • There are no parentheses left.
  • No further distribution is required.
This expression is as simple as it gets without further context, such as setting it equal to something for solving.

Remember, simplifying doesn鈥檛 change the value of the expression; it just makes it easier to view, compare, or solve.

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