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Chapter 4: Problem 60

Substract $$\begin{array}{r}-6 t^{3}+4 t^{2} \\\\-\left(8 t^{3}-6 t^{2}\right)\end{array}$$

Short Answer

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

Step by step solution

01

- Write the expression

Consider the given expression: \(-6t^3 + 4t^2 - (8t^3 - 6t^2)\).
02

- Distribute the negative sign

Remove the parentheses by distributing the negative sign: \(-6t^3 + 4t^2 - 8t^3 + 6t^2\).
03

- Combine like terms

Combine the like terms: \[(-6t^3 - 8t^3) + (4t^2 + 6t^2)\].
04

- Perform the addition and subtraction

Add or subtract the coefficients of the like terms: \[-14t^3 + 10t^2\].

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Key Concepts

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

Distributing negative sign
When encountering a subtraction problem with polynomials, the first task is to distribute the negative sign. This means we need to apply the negative sign to each term inside the parentheses. Think of it as multiplying each term by \(-1\).

For instance, in the given problem, \(- (8t^3 - 6t^2)\), you need to apply the negative sign to both \(8t^3\) and \(-6t^2\).

This transforms the expression from \( - (8t^3 - 6t^2) \) to \(-8t^3 + 6t^2\).

So, the polynomial \( -6t^3 + 4t^2 - (8t^3 - 6t^2) \) becomes \(-6t^3 + 4t^2 - 8t^3 + 6t^2\). This step ensures that all terms are correctly accounted for in the subtraction process.
Combining like terms
After distributing the negative sign, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power.

From the new expression \(-6t^3 + 4t^2 - 8t^3 + 6t^2\) look for terms with \(t^3\) and terms with \(t^2\).

In this expression, \(-6t^3\) and \(-8t^3\) are like terms because they both have \(t^3\). Similarly, \(4t^2\) and \(6t^2\) are like terms because they both have \(t^2\).

To combine them, simply add or subtract their coefficients:
  • Combine \(-6t^3\) and \(-8t^3\): the sum is \(-14t^3\)
  • Combine \(4t^2\) and \(6t^2\): the sum is \(10t^2\)
Combining like terms simplifies the polynomial to \(-14t^3 + 10t^2\).
Subtracting polynomials
Now, let's put it all together. Subtracting polynomials involves distributing the negative sign and then combining like terms, as we've discussed.

Given the original expression \(-6t^3 + 4t^2 - (8t^3 - 6t^2)\), we first distribute the negative sign:
  • Apply the negative sign to each term after the subtraction: \(-6t^3 + 4t^2 - 8t^3 + 6t^2\)
Then, we combine the like terms:
  • Combine \(-6t^3\) and \(-8t^3\) to get \(-14t^3\)
  • Combine \(4t^2\) and \(6t^2\) to get \(10t^2\)
This simplifies the given polynomial to \(-14t^3 + 10t^2\).

Always remember:
  • Distribute the negative sign carefully.
  • Combine only the like terms.
This method helps in accurately subtracting polynomials and avoiding common mistakes.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x+3)^{2}\\\ &\begin{array}{c|c|c|c|c|c} x & -5 & -4 & -3 & -2 & -1 \\ \hline y & & & & & \end{array} \end{aligned} $$

$$\text { Subtract the sum of } 9 t^{3}-3 t+8 \text { and } t^{2}-8 t+4 \text { from the sum of } 12 t+8 \text { and } t^{2}-10 t+3$$

Add or subtract as indicated. \((8 a b+2 a-3 b)-(6 a b-2 a+3 b)\)

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

If it costs \(\$ 15\) plus \(\$ 2\) per day to rent a chain saw, the binomial $$2 x+15$$ gives the cost in dollars to rent the chain saw for \(x\) days. Evaluate this binomial for \(x=6\). Use the result to fill in the blanks: If the saw is rented for_________ days, then the cost is _________ dollars.
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