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Chapter 5: Problem 28

Multiply. $$ -2 x^{3}\left(x^{2}-1\right) $$

Short Answer

Expert verified
-2x^5 + 2x^3

Step by step solution

01

- Distribute the term

To multiply \(-2x^3\) with the polynomial \(x^2 - 1\), distribute \(-2x^3\) to each term inside the parenthesis.
02

- Multiply \( -2x^3 \) with \( x^2 \)

Multiply \(-2x^3\) by \(x^2\): \(-2x^3 \times x^2 = -2x^{5}\).
03

- Multiply \( -2x^3 \) with \( -1 \)

Next, multiply \(-2x^3\) by \(-1\): \(-2x^3 \times -1 = 2x^3\).
04

- Combine the results

Combine the results from Steps 2 and 3: \(-2x^5 + 2x^3\).

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Key Concepts

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

Distributive Property
The distributive property is a key concept in algebra. It helps you multiply a single term by each term inside a set of parentheses.

In our given exercise \(-2x^3(x^2 - 1)\), we use the distributive property to distribute \(-2x^3\) to both \(x^2\) and \(-1\).

Here's how it works:
  • First, multiply \(-2x^3\) by \(x^2\).
    The result is \(-2x^3 \times x^2 = -2x^5\).
  • Next, multiply \(-2x^3\) by \(-1\).
    This gives \(-2x^3 \times -1 = 2x^3\).
By distributing the term correctly, you achieve \(-2x^5 + 2x^3\). The distributive property is essential because it simplifies complex expressions and makes multiplication straightforward.
Negative Exponent Multiplication
Multiplying terms with negative exponents can seem tricky, but it's just an extension of basic multiplication rules.

When dealing with exponents, remember these rules:
  • Exponents with the same base can be added together when multiplied.
    For example, \(a^m \times a^n = a^{(m+n)}\).
  • A negative sign in front of an exponent base flips the equation's direction as the exponent acts on multiplication.

In our exercise, consider the term multiplication: \(-2x^3 \times x^2\).
Here, you just need to add the exponents: \(3 + 2 = 5\). The result is \(-2x^5\).

Remember, negative signs outside the exponents affect the overall sign of the result, not the multiplication of the exponents themselves. Practicing with different exponents will make you comfortable with these operations.
Combining Like Terms
Combining like terms is an essential skill for simplifying expressions. Like terms have the same variable raised to the same power.

Let's see how this works in the context of our exercise. We end with the expression \(-2x^5 + 2x^3\). Here:
  • \(-2x^5\) and \(+2x^3\) are like terms, because both contain \('x'\) raised to some power.
  • When combining like terms, ensure exponents and bases match perfectly.

Combining terms helps you simplify your solution. In this exercise, there's no need to combine further since \(-2x^5\) and \(+2x^3\) are distinct terms.
Understanding and identifying like terms is crucial for more advanced algebra, making your math work clear and accurate.

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Most popular questions from this chapter

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=8 x^{3}+27, g(x)=2 x+3 $$

For each pair of functions \(f\) and \(g,\) determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{2 x}{x+1}\\\ &g(x)=2 x+5 \end{aligned} $$

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

Compare \(8 \times 10^{-90}\) and \(9 \times 10^{-91} .\) Which is the larger value? How much larger? Write scientific notation for the difference.

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 3^{5} \square 3^{4} $$
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