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Chapter 5: Problem 52

Subtract. $$ \begin{array}{r} {-6 t^{3}+4 t^{2}} \\ {8 t^{3}-6 t^{2}} \end{array} $$

Short Answer

Expert verified
-14t^3 + 10t^2

Step by step solution

01

Set Up Subtraction

Write the given polynomials one below the other with a subtraction sign between them. \[ (-6t^3 + 4t^2) - (8t^3 - 6t^2) \]
02

Distribute the Negative Sign

Distribute the negative sign to each term in the second polynomial: \[ -6t^3 + 4t^2 - 8t^3 + 6t^2 \]
03

Combine Like Terms

Combine the terms with the same degrees: \[ (-6t^3 - 8t^3) + (4t^2 + 6t^2) \]
04

Simplify Each Group

Perform the arithmetic operation within each group: \[ -14t^3 + 10t^2 \]

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Key Concepts

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

Combining Like Terms
When working with polynomials, 'combining like terms' is an essential skill to master.
Like terms are terms that have the same variable raised to the same power.
For example, in the expression \-6t^3 - 8t^3 + 4t^2 + 6t^2, \, -6t^3 and -8t^3 are like terms because they both contain \( t^3 \).
Likewise, 4t^2 and 6t^2 are like terms as both involve \( t^2 \).
To combine like terms, simply add or subtract their coefficients.
For example, combining -6t^3 and -8t^3 gives: \ -6t^3 - 8t^3 = -14t^3.
Similarly, combining 4t^2 and 6t^2 results in: \ 4t^2 + 6t^2 = 10t^2.
This simplifies your polynomial and makes it easier to work with.
Distributing Negative Signs
In polynomial subtraction, one key step is 'distributing negative signs'.
This means you'll change the signs of all the terms in the polynomial that comes after the subtraction sign.
This step ensures that you're correctly subtracting each term.
For example, in our problem: \ \ \ \((-6t^3 + 4t^2) - (8t^3 - 6t^2)\), we need to distribute the negative sign in front of the second polynomial:\ \[ -6t^3 + 4t^2 - (8t^3 - 6t^2) \ \ = -6t^3 + 4t^2 - 8t^3 + 6t^2 \]. Notice how each term's sign in \(8t^3 - 6t^2 \) is reversed.
This step is vital because it affects how you'll combine like terms.
Simplifying Expressions
Simplifying expressions is the final step in solving polynomial subtraction.
Once you've combined all like terms, your expression should be as simple as possible.
Let's consider our simplified groups: -14t^3 and 10t^2.
Our final polynomial is obtained by writing these simplified groups together: \ \ \(-14t^3 + 10t^2 \).
There's nothing more to combine or simplify here, so this is our final answer.
Always ensure that your final polynomial is in standard form, with the terms written in decreasing order of their degree.

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Most popular questions from this chapter

Assume that \(a\) is a number greater than 1 . Arrange the following terms in order from least to greatest: \(-(-a)^{3},-a^{3},(-a)^{4},-a^{4} .\) Explain how you decided on the order.

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Perform each division using the "long division" process. $$ \frac{3 k^{3}-4 k^{2}-6 k+10}{k-2} $$
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