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Chapter 5: Problem 113

Simplify. \(4 t^{4}+3 t^{2}+8 t-\left(3 t^{4}+9 t^{2}+8 t\right)\)

Short Answer

Expert verified
The simplified expression is t^4 - 6t^2.

Step by step solution

01

- Distribute the Negative Sign

Begin by distributing the negative sign across the terms inside the parenthesis. This means changing the sign of each term within the parenthesis: ewline 4t^4 + 3t^2 + 8t - (3t^4 + 9t^2 + 8t) = 4t^4 + 3t^2 + 8t - 3t^4 - 9t^2 - 8t
02

- Combine Like Terms

Next, combine the like terms: terms with the same variable raised to the same power. Group the like terms together: ewline (4t^4 - 3t^4) + (3t^2 - 9t^2) + (8t - 8t)
03

- Perform Arithmetic Operations

Finally, simplify each group of like terms by performing the necessary arithmetic operations: ewline (4 - 3)t^4 + (3 - 9)t^2 + (8 - 8)t which simplifies to t^4 - 6t^2 + 0 Since the term with coefficient 0 disappears, we get t^4 - 6t^2

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Key Concepts

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

Distributing the Negative Sign
The first step in many algebraic simplifications involves distributing a negative sign. This means you change the sign of every term within the parentheses.
For example, in our exercise, we have:
\(4t^4 + 3t^2 + 8t - (3t^4 + 9t^2 + 8t)\)
Here, the negative sign before the parenthesis needs to be distributed to each term inside the parenthesis. Changing the signs, we get:
\(4t^4 + 3t^2 + 8t - 3t^4 - 9t^2 - 8t\).
This step is crucial as it prepares the expression for easier simplification.
Failure to properly distribute the negative sign will lead to incorrect results in your final answer.
Combining Like Terms
After distributing the negative sign, the next step in polynomial simplification is to combine like terms.
Like terms are terms that have the same variable raised to the same power.
In this case, we identify and group them together:
\((4t^4 - 3t^4) + (3t^2 - 9t^2) + (8t - 8t)\).
  • \(4t^4\) and \(-3t^4\) are like terms because they both contain \(t^4\).

  • \(3t^2\) and \(-9t^2\) are like terms because they both contain \(t^2\).

  • \(8t\) and \(-8t\) are like terms because they both contain \(t\).

By grouping and then combining these like terms, you simplify the expression further.
Polynomial Simplification
The final step in simplifying a polynomial involves performing the arithmetic operations on the grouped like terms.
In our example, we have:
\((4 - 3)t^4 + (3 - 9)t^2 + (8 - 8)t\)
Let's break it down:
\((4 - 3)t^4\) simplifies to \(1t^4\) or simply \(t^4\)
\((3 - 9)t^2\) simplifies to \(-6t^2\)
\((8 - 8)t\) simplifies to \(0t\), which we can drop because it equals zero.
Putting it all together, the simplified polynomial is:
\(t^4 - 6t^2\)
By following these three steps consistently, you can simplify even more complex polynomials with ease.

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