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Chapter 5: Problem 48

Subtract. $$ \left.7+t-5 t^{2}+2 t^{3}\right)-\left(1+2 t-4 t^{2}+5 t^{3}\right) $$

Short Answer

Expert verified
6 - t - t^2 - 3t^3

Step by step solution

01

Write down the expressions inside parentheses

The original problem is to subtract the following two polynomials: (7 + t - 5t^2 + 2t^3) - (1 + 2t - 4t^2 + 5t^3).
02

Distribute the negative sign

Distribute the negative sign to each term in the second polynomial: (7 + t - 5t^2 + 2t^3) - 1 - 2t + 4t^2 - 5t^3.
03

Combine like terms

Combine the terms with the same degree: Constant terms: 7 - 1 = 6 t-terms: t - 2t = -t t^2-terms: -5t^2 + 4t^2 = -t^2 t^3-terms: 2t^3 - 5t^3 = -3t^3.
04

Write the simplified polynomial

Combine all the like terms to write the simplified polynomial: 6 - t - t^2 - 3t^3.

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Key Concepts

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

Combine Like Terms
When working with polynomials, an important step is to combine like terms. Like terms are terms that have the same variable raised to the same power. This makes simplifying the polynomial easier.
For example, in the polynomial given, we have terms such as:
  • Constants: 7 and 1
  • Linear terms: t and 2t
  • Quadratic terms: -5t^2 and -4t^2
  • Cubic terms: 2t^3 and 5t^3
When we combine these like terms, we add or subtract their coefficients while keeping the variable part the same. For constants, we subtract as follows: 7 - 1 = 6. For linear terms, it is: t - 2t = -t. Quadratic terms combine as -5t^2 + 4t^2 = -t^2. Lastly, cubic terms become 2t^3 - 5t^3 = -3t^3.
Distribute Negative Sign
Another critical step in polynomial subtraction is distributing the negative sign. This negative sign affects every term within the polynomial it is in front of. This means we flip the sign of each term inside the second parentheses.

For example, in the provided exercise:
  • First polynomial: (7 + t - 5t^2 + 2t^3)
  • Second polynomial with distributed negative sign: - (1 + 2t - 4t^2 + 5t^3) = -1 - 2t + 4t^2 - 5t^3
Applying the negative sign correctly is essential. If not done accurately, it can lead to wrong results when combining like terms and simplifying the polynomial expression.
Simplify Polynomial
After distributing the negative sign and combining like terms, the final step is to simplify the polynomial. This involves gathering all like terms and writing them as a single polynomial expression.
In the given exercise:
The expression becomes:
  • Constant terms: 6
  • Linear terms: -t
  • Quadratic terms: -t^2
  • Cubic terms: -3t^3
Putting these together, we get the simplified polynomial: 6 - t - t^2 - 3t^3. Simplifying makes the polynomial easier to work with and understand, which is crucial for future calculations or solutions.

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Most popular questions from this chapter

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D4 contains the formula $$ =2 * \mathrm{A} 4+3 * \mathrm{B} 4 $$ What is the value in \(\mathrm{D} 4\) if the value in \(\mathrm{A} 4\) is 5 and the value in \(\mathrm{B} 4\) is \(10^{2}?\)

Simplify. \(0.1 n^{2}+5-\left(-0.3 n^{2}+n-6\right)\)

Solve. Write the answers using scientific notation. Without performing actual computations, explain why \(3^{-29}\) is smaller than \(2^{-29}\)

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=3 x-2\\\ &g(x)=2 x-8 \end{aligned} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3+(-11) $$
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