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Chapter 3: Problem 63

Perform the indicated operations and simplify as completely as possible. $$x^{2}\left(x^{3}-4 x\right)-\left(2 x^{5}-4 x^{3}\right)$$

Short Answer

Expert verified
\(-x^5\)

Step by step solution

01

- Distribute the exponent in the first expression

Multiply each term inside the first parenthesis by the term outside: \(x^2 \cdot (x^3 - 4x)\). This results in: \(x^2 \cdot x^3 - x^2 \cdot 4x = x^5 - 4x^3\).
02

- Recognize the subtraction

Subtract the second expression \((2x^5 - 4x^3)\) from the simplified first expression \((x^5 - 4x^3)\).
03

- Perform the subtraction

Combine like terms by subtracting the coefficients: \(x^5 - 4x^3 - 2x^5 + 4x^3\).
04

- Simplify the expression

Combine like terms to simplify further: \( (1x^5 - 2x^5) + (-4x^3 + 4x^3) = -x^5 + 0\).
05

- Write the final simplified expression

The simplified result is: \(-x^5\).

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Key Concepts

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

Distributing Exponents
When you see an expression like \(x^2(x^3 - 4x)\), you need to distribute \(x^2\) to each term inside the parentheses. In this case, distribute \(x^2\) to both \(x^3\) and \(4x\).

Here鈥檚 what you get:
  • \(x^2 \times x^3 = x^{2+3} = x^5\)
  • \(x^2 \times (-4x) = -4x^{2+1} = -4x^3\)
So, from \(x^2(x^3 - 4x)\), you get \(x^5 - 4x^3\).

Remember, when multiplying terms with the same base, you add their exponents. That鈥檚 why \(x^2 \times x^3\) becomes \(x^{2+3} = x^5\). Also, distribute the negative sign appropriately to get the correct result.
Combining Like Terms
Like terms are terms that have the same variable raised to the same power. For example, \(x^3\) and \(-4x^3\) are like terms, but \(x^3\) and \(x^5\) are not. In the expression \(x^5 - 4x^3 - (2x^5 - 4x^3)\), we need to identify and combine like terms.

First, simplify each part within the parentheses:
  • The first expression is already simplified to \(x^5 - 4x^3\).
  • The second expression remains \(2x^5 - 4x^3\).
Now, combine the corresponding like terms:
  • For \(x^5\): Combine \(x^5\) from the first group and \(-2x^5\) from the second group to get \((1-2)x^5 = -x^5\).
  • For \(x^3\): Combine \(-4x^3\) from both groups to get \(-4x^3 + 4x^3 = 0\).
This step involves straightforward arithmetic with coefficients and makes combining like terms easy.
Simplifying Expressions
Simplifying an expression means reducing it to its simplest form. After distributing and combining like terms, let's simplify further.

Given: \(x^5 - 4x^3 - (2x^5 - 4x^3)\).

From the previous step, we know our terms are \(x^5 - 4x^3 - 2x^5 + 4x^3\).

Now, combine the like terms:
  • \(x^5\) and \(-2x^5\): These terms simplify to give \(-x^5\)
  • \(-4x^3\) and \(4x^3\): These cancel each other out, giving \(0\).
So, the simplified form of the expression is \(-x^5\).

This step-by-step reduction helps ensure you arrive at the simplest form of a polynomial.
Subtraction of Polynomials
Subtracting polynomials involve changing the sign of each term in the polynomial being subtracted and then combining like terms. Starting with \((x^5 - 4x^3) - (2x^5 - 4x^3)\), distribute the negative sign in the second polynomial:

The expression becomes \(x^5 - 4x^3 - 2x^5 + 4x^3\).

Each term in \(2x^5 - 4x^3\) is taken to the opposite sign:
  • \(2x^5\) becomes \(-2x^5\).
  • \(-4x^3\) becomes \(4x^3\).
Now combine like terms:
  • \(x^5 - 2x^5 + (-4x^3 + 4x^3)\).
  • Simplify by performing the arithmetic: \(x^5 - 2x^5 = -x^5\) and \(-4x^3 + 4x^3 = 0\).
The final result is \(-x^5\).

Remember to change the signs when subtracting, then combine like terms to reach the simplified form.

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